Stability of an age specific population with density dependent fertility

A nonlinear version of the Lotka-Sharpe model of population growth is considered in which the age specific fertility is a function of the population size. The stability of an equilibrium population distribution is investigated with respect to both global and local perturbations. Sufficient conditions for such stability are presented, as are estimates for the rate of return of the population to the equilibrium configuration. Particular attention is paid to those situations in which the age dependent stability criteria coincide with those of age independent models.     

具有年龄结构的非线性种群发展方程

本文研究了笔者与马知恩教授所提出的一类具有年龄结构的种群发展模型,该模型较完满地解决了古典的Shape-Lotka方程所存在的忽略了孵化期的问题。文中对非线性系统解的适定性、种群发展的有界性和平衡分布作了讨论,利用算子半群理论研究了解的渐近性态,并考虑了一类特殊模型;对结果给予了生物解释。     

Female dominant age-dependent deterministic population dynamics

Summary In this paper the study of a nonlinear deterministic model of an age/sex differentiated population is started with a proof of existence and uniqueness of solution of the governing equation. We also obtain time dependent upper bounds to the male and female populations as well as necessary and sufficient conditions for equilibrium.     

Interannual Variations of the Body Length in Sockeye Salmon Oncorhynchus nerka Smolts of Lake Kuril'skoe in Relation to External Factors

A nonlinear regression model of body length dynamics was constructed for age smolts greater than two years old of the sockeye salmon Oncorhynchus nerka of Lake Kuril'skoe (Kamchatka). The analysis of variance showed that about 93% of the variation of body length is accounted for by the fish population density and water temperature. The reasons for the increase in the body length of smolts after volcanic ashfall are discussed.     

Diagnosing Aging: I. Problem of Reliability of Linear Regression Models of Biological Age

The problem of the reliability of linear regression models of biological age assessment was studied using an experimental population of patients of a geroprophylactic center. The main factors of the model quality (interpopulation difference, method of approximation of biological age, and methods of approximation of statistical significance of parameters of biological age models) were tested. New equations were derived for calculating biological age. All parameters of these equations meet the requirements of significance. It was shown that if the nonlinear character of age dynamics of biological markers of aging and the statistical significance of model parameter estimates are taken into account, the model of biological age is substantially simplified and its reliability increases.     

On synchronization in semelparous populations

Synchronization, i.e., convergence towards a dynamical state where the whole population is in one age class, is a characteristic feature of some population models with semelparity. We prove some rigorous results on this, for a simple class of nonlinear one- population models with age structure and semelparity: (i) the survival probabilities are assumed constant, and (ii) only the last age class is reproducing (semelparity), with fecundity decreasing with total population. For this model we prove: (a) The synchronized, or Single Year Class (SYC), dynamical state is always attracting. (b) The coexistence equilibrium is often unstable; we state and prove simple results on this. (c) We describe dynamical states with some, but not all, age classes populated, which we call Multiple Year Class (MYC) patterns, and we prove results extending (a) and (b) into these patterns.Acknowledgement Boris Kruglikov contributed the nonlinear part of the formulation as well as the proof of Theorem 1. The authors are grateful for critical and constructive comments by N. Davydova and O. Diekmann. E.M. is also grateful for discussions with Marius Overholt concerning problems of proving Theorem 2.     

大足鼠种群动态的非线性模型及逐步回归分析

本文提出了一种从原始数据中获取关于种群动态机制的基本信息的,以非线性模型为基础的逐步回归逐步预测的种群动态分析法。并以作者１９８９－１９９１年间于四川邛崃定点进行示志重捕以及夹捕解剖所得的关于农田大足鼠种群生态学特征的数据为例,对大足鼠和种群密度与年龄组成,各年龄组性比、繁殖特征以及气象等因子的关系进行了初步的分析,定量地确定了当前的各生态因子对未来的大足鼠种群密度的作用,并对数量了初步的分析的讨     

Periodic forcing of microbial cultures: a model for induction synchrony

Periodic environmental shifts have been used to induce synchrony in many different microbial populations. In this article, the induction synchrony phenomenon is analyzed using an age distribution model in which the age at which the cells divide is subjected to periodic forcing. It is found that synchrony will occur whenever the period of the forcing lies in the interval between the youngest and the oldest division age that occur in the population during the forcing. The analysis also predicts that under certain conditions it should be possible to obtain a multimodal synchrony in which cells in the population are distributed among a set of discrete, synchronized cell lines. The behavior of the age distribution when the conditions for synchrony are not satisfied is briefly explored. It is found that the age distribution model is able to exhibit a very rich spectrum of possible dynamic behavior. Many of the phenomena observed can be thought of in terms that are familiar from nonlinear analysis, such as stable and unstable limit cycles, period doubling, halving, and chaos. The richness of dynamic behavior opens the possibility that environmental shifts or periodic forcing could be used as a powerful tool in discriminating models of microbial kinetics and cell cycle control.     

Further analysis of Clark's delayed recruitment model

I explore the nonlinear behavior of a model in which the number of adults in each year is the sum of recruitment (which depends nonlinearly on adult abundanceT years in the past) and a constant fraction (survival) of adults in the previous year. Adding even a small amount of age structure to the semelparous version of this model (by increasing adult survival from zero) is stabilizing in that: (1) it shifts the value of slope at which the linearized model becomes unstable about the equilibrium to a lower value; and (2) it stretches the pattern of period doubling bifurcations so that bifurcations occur at much lower values of the slope. For the iteroparous case with a maturation and survival pattern reasonably typical of long-lived organisms, the period of cycles or quasi-cycles produced increases continuously as the slope of the stock-recruitment function decreases. The possibility of arbitrarily long cycles is not predicted by the linear theory, and has important practical implictions for analyses of cyclic populations. Both truncation of the age structure and an upper limit on recruitment seem to remove this gradual increase in period. However, the former can give rise to doubling of the period. Although the nonlinear behavior is not analysed in detail, a qualitative interpretation of the behavior of this model in terms of population inertia seems to explain the behavior observed in these numerical simulations.     

Maximum sustainable yield of a density dependent sex-age-structured population model

The maximum sustainable yield of an age-structured, density dependent, sex-differentiated population is investigated. The model is based on the nonlinear version of the McKendrick [11] model introduced by Gurtin and MacCamy [8], modified to include sex-differentiated dynamics. It is determined that the maximum sustainable yield is attainable by an age specific harvesting policy in which the number of harvesting ages for males and the harvesting ages for females total at most five.     

Risk spreading as an adaptive strategy in iteroparous life histories

It has long been conjectured, though without satisfactory proof, that life tables with a long reproductive span are advantageous in an environment where fecundity or immature survival rates fluctuate randomly. In the present analysis we recast the nonlinear Leslie matrix problem as an autoregressive time series model for the birth rate, with random addition and removal of newborn. This transformation renders the model linear with respect to the environmental variation, allowing ready solution for the ultimate population size and for the conditions resulting in stationarity of the population distribution. We show that for life tables where the fecundities of all adult age classes are the same (no restrictions are put on the survivorship schedule, or on the age at first reproduction), and where density dependence operates via total adult density, the realized growth rate is less than the growth rate calculated from the mean Leslie matrix associated with the population's growth history. The degree of the discrepancy increases with the environmental variability, and decreases with iteroparity, thus completing a proof which confirms the correctness of the initial conjecture for a class of biologically reasonable lifetable models.     

Aging and nonlinear heart rate control in a healthy population

In recent years more studies are using nonlinear dynamics to describe cardiovascular control. Because of the large dispersion of physiological data, it is important to have large studies with both male and female participants to establish a range of physiological healthy values. This study investigated the effect of gender and age on nonlinear indexes. Nonlinear scaling properties were studied by using 1/f slope (where f is frequency), fractal dimension, and detrended fluctuation analysis short- and long-term correlations (DFAalpha(1) and DFAalpha(2), respectively). Nonlinear complexity was described with correlation dimension (CD), Lyapunov exponent (LE), and approximate entropy (ApEn). The population consisted of 135 women and 141 men (age, 18-71 yr). Twenty-four hour ECG recordings were obtained by using Holter monitoring. The recordings were split into daytime (8 AM-9 PM) and nighttime (11 PM-6 AM). A day-night variation was present in all nonlinear heart rate variability (HRV) indexes, except for the CD in the female population. During the night the percentage of CD values of surrogate data files differing from the CD value of the original data increased. All nonlinear indexes were significantly correlated with age. Deeper analysis per age category of 10 yr showed a stabilization in the age decline of the fractal dimension and ApEn at the age of > or =40 yr. The vagal pathways seemed to be more involved in the generation of nonlinear fluctuations. Higher nonlinear behavior was evident during the night. No clear difference between men and women was found in the nonlinear indexes. Nonlinear indexes decline with age. This can be related to the concept of decreasing autonomic modulation with advancing age.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

A mathematical model for the phase of sexual reproduction in monogonont rotifers

Recently, the optimal sex allocation in monogonont rotifers is studied in [1], and, as a closely related question, the relative frequencies of the relevant types of mictic females. The authors focus on the evolution of the age at which young mictic females lose their fertilization susceptibility and they address the threshold age of fertilization that maximizes resting egg production. Assuming that a stationary population is achieved, with stable age distribution, they obtain their results, without knowing the stationary population. Our aim is to study this problem in the framework of the theory of nonlinear age-dependent population dynamics developed by G. F. Webb in [13], which is more appropriate from the mathematical point of view and permits to us to obtain analytically the stationary population and consequently it is analytically shown that a threshold age of fertilization equal to the age of maturation is not an ESS, despite the fact that then the production of resting eggs is maximum, which has been obtained by simulation in [1]. Received: 30 June 1998 / Revised version: 17 May 1999 / Published on line: 5 May 2000     

The optimal sex ratio for age-structured populations

A new nonlinear age-dependent model for age-structured sexual populations is introduced, based on two assumptions: (1) the birth function depends on the ages of the two parents; and (2) the death functions of the two sexes are composed of two types of additive terms depending on age and sex and on time evolution of population densities, respectively. Formal arguments are given that suggest that time-persistent age profiles may exist and that the intrinsic rate of growth for the two sexes is the same. If the ratio between the number of newborn females and the number of newborn males is equal to the square root of the ratio of the corresponding per capita birth rates, then the intrinsic rate of growth has an optimal value. The optimal sex ratio for the whole population is equal to the reciprocal value of the sex ratio at birth.     

Persistent distribution functions for the dispersion of structured populations

This paper primarily expounds upon the problem of persistent age-state distribution functions for the dispersion of structured populations. A general model is introduced, based on the following assumptions: 1) the state of an individual of age a is characterized by a set of random variables X₁, X₂,…, X_Q (weight, size, etc.) obeying a phenomenological master equation; 2) the birth function λ depends on the age a' of parents and on the state variables X₁,…, X_Q of the newborns; 3) the mortality function is composed of two additive terms—the first contribution depends only on age while the second contribution depends on the total population density; 4) the population diffuses to avoid crowding. These hypotheses define a nonlinear population model for which time- and space-persistent age-state distribution functions eventually may occur even if the total population density is time- and space-dependent. A biological interpretation of the main results is given in terms of the distribution function of the state vector at birth. In the last part of the paper a generalized model is presented, assuming that the behavior of an individual is described by a system of age-dependent master equations [29].     

Equilibria in structured populations

The existence of a stable positive equilibrium state for the density of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany     

State-vector models of the cell cycle 1. Parametrization and fits to labeled mitoses data

In this paper we study the Hahn model of the cell cycle from the point of view that a cell population's age distribution is more relevant to labeled mitoses data than is the distribution of its transit times.Closed-form relationships are derived between the transition probabilities of the Hahn model and the transit time of the mean of a cohort of labeled cells (with the variance of their transit time through mitosis). Constraints result which define the acceptable values for the number of ages in the state vector and the length of the time step (rarely does the dimension of the state vector equal the number of time steps in the generation time).A generalization to distinct probabilities for G₁, S and G₂M is presented, and the automatic fitting of fraction-labeled mitoses (FLM) data is described. The doubling time of the population is used to define the daughter factor, via the largest eigenvalue of the state transition matrix. The performance of the generalized Hahn model is compared to that of other commonly used fitting methods using two sets of FLM data from the literature. The synthesis of continuous labeling curves is discussed as an independent check of the parametrization. Based on the stable age distribution resulting from fits to experimental FLM data, it is shown that a nonlinear relationship exists between biological age and time.