A dynamic multistate model of robustness and frailty

Dynamic multistate models can show realistic population dynamics over time, model complex cycles, and encompass the life history of a cohort. This paper uses a recently developed approach to obtain the analytic solution of a time‐dependent multidimensional differential equation. The illustrative robust/frail model presented shows that the assumption of fixed individual frailty can be abandoned in a two living state model that allows transitions between health statuses and nonproportional hazards of death.     

Macro‐demographic effects of the transition to adulthood: Multistate stable population theory and an application to Italy

We exploit a multistate generalisation of a classical, one‐sex, stable population model to evaluate structural and long‐term effects of changes in the attainment of adulthood. The demographic framework that inspired this paper is provided by Italy, where a strong delay in the transition to adulthood and union formation has been observed over the last several decades. Italy has also experienced very low fertility levels, and the subsequent ageing problems have become of primary concern. We first discuss a theoretical framework based on the model developed by Inaba (1995) and then include the process of transition to adulthood. We consider explicitly some specifications of the general model, and we present two distinct empirical applications, one using macrosimulation and the other one using a linear approximation. Our principal aim is to evaluate the impact of the delay in the attainment of adulthood on reproduction and on the age structure of the population.     

The multistate life table with duration‐dependence

A method for generalizing the multistate, or increment‐decrement, life table to include rates which depend upon duration of exposure to risk, as well as upon age, is proposed. The method is built upon the linear approximation, called the linear integration hypothesis, developed mainly by Rogers and his colleagues. Although the use of rates indexed by duration categories leads to a substantial increase in the state space of the model, it is possible to arrange the rates in such a way that matrices to be inverted are no larger than those encountered in the usual multistate life table. In the more general approach it is possible to derive several new summary indices of the life‐table cohort's history, such as the mean and median time in current status, at any age. The method is illustrated using a simple four‐state marital‐status model which has appeared often in the literature; here, rates of divorce and widowhood vary by duration of marriage as well as age.     

Asymptotic properties of the inhomogeneous Lotka‐Von Foerster system

In this paper, we investigate an extension of the multistate stable population model, which makes allowances for migration. The model is formulated as an inhomogeneous system of first order partial differential equations with integral boundary conditions. First, we construct its classical solution. Next, we reformulate the system as an abstract inhomogeneous Cauchy problem on a Banach space, and give its mild solution by using the population semigroup. Our main purpose is to investigate the asymptotic behavior of the mild solution.     

Stable growth in native‐dependent multistate population dynamics

Users of multistate life tables and projections have recognized that the Markovian assumptions underlying such models are unduly restrictive and should be relaxed whenever data permit. Efforts to include the influences of previous occupancies have included the incorporation of place‐of‐birth dependence. This paper addresses the stable growth properties of such generalized multistate models. It shows how place‐of‐birth‐specific stable growth measures can be calculated without projection simply by solving the characteristic equation. An example using Canadian data illustrates the argument.     

Population growth with variable family size

The Sharpe‐Lotka continuous time deterministic model of population growth is developed to take account of some possible forms of mother‐daughter fertility association, characterised here by a bivariable measure, A. This leads to a linear double integral equation for which, subject to certain conditions, a finite time solution can be found by Laplace transform methods and thus also model specific results relating the intergenerational fertility effect to the long term population growth rate and magnitude are established. The quantitative implications of the theory are illustrated by a consideration of a general bilinear form of A and in this context numerical results illustrating the finite time growth and also the long term distribution of fertility levels in the stable female population are obtained. In particular, it is shown that different fertility specific subpopulations can coexist indefinitely.     

Persistent age distributions for an age‐structured two‐sex population model*

In this paper we formulate an age‐structured two‐sex population model which takes into account a monogamous marriage rule and the duration of marriage. We are mainly concerned with the existence of exponential solutions with a persistent age distribution. First we provide a semigroup method to deal with the time‐evolution problem of our two‐sex population model. Next, by constructing a fixed point mapping, we prove the existence of exponential solutions under homogeneity conditions.     

A stochastic version of the malthusian trap model: Consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross‐sectional correlation between these two growth rates under two additional assumptions: i) the relations in the model at national levels include country‐specific and time‐invariant random components, and ii) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these two assumptions can explain near‐zero correlations between the two growth rates even if there exist a strongly negative effect of population growth on economic growth. However it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

A semigroup approach to the strong ergodic theorem of the multistate stable population process

In this paper we first formulate the dynamics of multistate stable population processes as a partial differential equation. Next, we rewrite this equation as an abstract differential equation in a Banach space, and solve it by using the theory of strongly continuous semigroups of bounded linear operators. Subsequently, we investigate the asymptotic behavior of this semigroup to show the strong ergodic theorem which states that there exists a stable distribution independent of the initial distribution. Finally, we introduce the dual problem in order to obtain a logical definition for the reproductive value and we discuss its applications.     

An age‐structured model of population dynamics with dominant ages,delayed behavior,and oscillations

An age‐structured model of population dynamics with age‐dominance is proposed and analyzed. Existence and uniqueness of solutions are established as well as the uniqueness and local asymptotic stability of steady‐states. Conditions for convergence to or oscillation about the steady‐state are specified in some cases.     

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.     

Induced population growth and induced technological progress: Their interaction in the accelerating stage

A simple model of Malthusian population growth combined with population‐induced technological progress generates accelerating growth. The model may be relevant for a first stage of growth in which natural resource limitations can be overcome through technological progress; it is not applicable to a later stage in which resource constraints are more resistant. Parameter values are roughly inferred from historical experience. Exogenously more rapid population growth initially depresses income, perhaps for up to several centuries, then raises it without limit. More rapid population growth is desirable only when the social discount rate is less than the ratio of the parameters for induced technical progress and static diminishing returns. Imposed population fluctuations cause inverse movements in incomes, so that induced progress is very difficult to detect empirically even for population fluctuations up to 500 years.     

Populations with constant immigration

We consider a Leslie‐type model of a one‐sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie‐type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigratioa     

Turbulent dynamics in a Xviith century population

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

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

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

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

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

Conditions determining the local stability of populations

The conditions that determine the local stability classification of an equilibrium population configuration are analyzed. The population investigated is age‐structured and density‐dependent, where density is determined by an age‐weighted population size. Two demographic parameters are introduced: the marginal birth rate and marginal death rate, which describe the marginal density‐dependence of the birth and death rates of the equilibrium population. Certain necessary and/or sufficient conditions determining stability are developed, most of them involving the net reproduction rate of the population, and examples illustrating these conditions are presented.     

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.     

The mathematics of sex and marriage,revisited

We analyze the problem of modeling marriages in a two‐sex model of population dynamics. We first deal with the problem of incomplete and inconsistent census data and then use a simulator to compare the performance of a variety of marriage functions in modeling births and couples during the ten‐year period between consecutive U.S. censuses. Unlike most empirical methods for comparing marriage functions based on goodness of fit, the differences in the projections of the various functions in our method are of the same magnitude (or even smaller) than the errors between the projected and real data. We observe that for the population of the United States, the harmonic mean function frequently found and used in the literature is a quite poor performer when compared with many other functions in the family we use.     

Pre‐procreative ages in population stability and cyclicity

Age‐specific models of population renewal (with and without feedback) which imply convergence to a stable state for some levels of fertility or feedback may imply the presence of long‐term cycling around a constant or exponentially changing equilibrium for other levels of fertility or feedback. The switch from one regime to the other is a “bifurcation.”; The conditions for bifurcation involve the roots of an analogue of Lotka's Equation.

Typically bifurcation is induced by raising the strength of feedback or the level of fertility. It has been known since the early 1980s, however, that this is sometimes impossible. It is sometimes impossible even with the linear renewal equation itself and with the most basic of non‐linear models, Lee's cohort feedback model.

Here it is proved that this typical route to bifurcation does not fail for these basic models in the presence of a condition which always holds for realistic applications with higher organisms: the existence of a span of ages before the onset of fertility.

Specifically, a strictly positive lower bound on ages of procreation is proved to be sufficient to guarantee the existence of a rescaling of Lotka's Equation for which the real part of some complex root vanishes. This result holds for absolutely Lebesgue‐integrable (signed) net maternity functions on the positive real line and for absolutely summable (signed) net maternities on the positive integers.

It follows that Coale's rescaling device for the analysis of approach to stability in stable population theory can be implemented for all realistic human net maternity schedules. It also follows that the many special cases of the cohort feedback model throughout population biology will all generate persistent cycling instead of stability if feedback is sufficiently strong.     

Three strikes and you're out: Demographic analysis of mandatory prison sentencing

Much of the debate about the costs and benefits of “three-strikes” laws for repeat felony offenders is implicitly demographic, relying on unexamined assumptions about prison population dynamics. However, even state-of-the-art analysis has omitted important demographic details. We construct a multistate life-table model of population flows to and from prisons, incorporating age-specific transition rates estimated from administrative data from Florida. We use the multistate life-table model to investigate patterns of prison population growth and aging under many variants of three-strikes laws. Our analysis allows us to quantify these demographic changes and suggests that the aging of prison populations under three-strikes policies will Significantly undermine their long-run effectiveness.     

Slow‐fast dynamics in a model of population and resource growth

Models of the interaction of population, the economy, and the environment often contain nonlinear functional relationships and variables that move at different speeds. These properties foster apparent unpredictabilities in system behaviour. Using a simple deterministic model of demographic, economic and environmental interactions we illustrate the usefulness of geometric singular perturbation theory in environmental population economics. In contrast to local stability analysis, the theory of slow‐fast dynamics helps to gain new insights into the global behaviour of the system. In particular, the knowledge of the basins of attraction of the stationary states enables one to determine the regions of sustainable future paths of resources and population.