On the history of populations governed by a Leslie matrix

Even though the Leslie matrix is usually singular, there is a subspace on which is has an inverse. In addition, there is a projection into that subspace which preserves certain age classes. These two facts are combined to provide a model for the history of a population whose future is predicted by a Leslie matrix. It has the advantage of being composed of easily calculated matrices. The relation of this model to a backward projection method of Greville and Keyfitz is discussed and some other backward projection functions are proposed.     

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. The level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.     

Compartmental models and their application : Keith Godfrey,Academic Press,London, 1983, xiv+293 pp., $50.00

For the Leslie-matrix population projection model, a simple auxiliary quantity, termed the single-year reduced growth factor, is introduced and used to deduce bounds for the growth factor of the model, i.e. the dominant eigenvalue of the Leslie matrix involved. Extension to the case of internal survival in the age groups and reformulation of parameter sensitivity are briefly discussed.     

The structure and dynamics of woodreed Calamagrostis canescens population: a modelling approach

A scale of ontogenetic states has been developed for woodreed Calamagrostis canescens, a perennial species dominating the grass layer of fell forest areas. The population structure is considered as a set of age-stage groups of individuals differing both in the ontogenetic stage and the chronological age measured in years. to describe the dynamics through years a special kind of matrix formalism has been proposed which is reducible neither to the classic Leslie matrix for an age-structured population, nor to the well-known Lefkovitch matrix for a stage-structured one, and which does not suffer from excessiveness of the "two-dimensional" representation for the structure implying the projection matrix of a block pattern. It has been shown however that the protection matrix corresponding to C. canescens life-history graph embodies the canonical features of matrix formalism for structured population dynamics, such as the exponential population growth or decline, the convergence to a stable equilibrium structure, the calculable indicator of growth/decline/equilibrium (i.e., a measure of the population reproductive potential) as well as possibility to determine the relative reproductive value of each group. On the other hand, "left-sidedness of the age spectrum", a property that is often observed in real populations and is inherent in Leslie models of growing populations, may fail in the age-stage-structured model. The aggregation of age-stage groups into the age classes is possible only under special strict relationship among the age-stage-specific vital rates of the population. The both circumstances serve a methodical indication that an additional dimension such as the stages, for example, ought to be introduced into the age structure of the model population.     

Computerization of a population projection model based on generalized age-dependent branching processes: a connection with the Leslie matrix

The population projection model based on generalized age-dependent branching processes developed by Mode and Busby (1981) involves the solution of a large number of renewal type equations. It is shown that these equations may be solved recursively. Such a solution has two implications. One is that the projection model may be very efficiently computerized. Second, the recursive algorithm developed has striking similarities to two traditional methods of population projection used by demographers: the Leslie matrix and cohort component methods. The results presented here associate traditional projection techniques with the theory of age-dependent branching processes.     

If a population crashes in prehistory, and there is no paleodemographer there to hear it, does it make a sound?

Catastrophic episodes (e.g., epidemics, natural disasters) strike with only limited regard for age. A large percentage of catastrophic mortality in a population can lead to a death distribution that resembles the living distribution, which includes greater numbers of older children, adolescents, and young adults than typical mortality profiles. This paper examines both the population implications of a large catastrophic mortality event, based on the Black Death as it ravaged medieval Europe, and its long-term effects on age-at-death distributions. An increased prevalence of epidemic disease is a common feature of reconstructions of the shift to agriculture and the rise of urban centers. The model begins with a hypothetical Medieval living population. This population is stable and characterized by slow growth. It has fertility and mortality rates consistent with a natural-fertility, agrarian population. The effects of catastrophic episodes are simulated by projecting the model population and subjecting it to one large (30% mortality) catastrophic episode as part of a 100-year population projection. A pair of Leslie matrices forms the basis of the projection. The catastrophic episode has important, long-term effects on both the living population and the cumulative distribution of death. The living population fails to recover from plague losses; at the end of the projection, population is still less than 75% its pre-plague level. The age-at-death distribution takes on the juvenile-young adult-heavy profile characteristic of many archaeological samples. The cumulative death profile based on the projection differs from that produced by the stable model significantly (P < 0.05) for 25-50 years after the plague episode, depending on sample size.     

Population dynamics in variable environments I. Long-run growth rates and extinction

A model of population growth is studied in which the Leslie matrix for each time interval is chosen according to a Markov process. It is shown analytically that the distribution of total population number is lognormal at long times. Measures of population growth are compared and it is shown that a mean logarithmic growth rate and a logarithmic variance effectively describe growth and extinction at long times. Numerical simulations are used to explore the convergence to lognormality and the effects of environmental variance and autocorrelation. The results given apply to other geometric growth models which involve nonnegative growth matrices.     

Matrix population model with density-dependent recruitment for assessment of age-structured wildlife populations

A logistic density-dependent matrix model is developed in which the matrices contain only parameters and recruitment is a function of adult population density. The model was applied to simulate introductions of white-tailed deer into an area; the fitted model predicted a carrying capacity of 215 deer, which was close to the observed carrying capacity of 220 deer. The rate of population increase depends on the dominant eigenvalue of the Leslie matrix, and the age structure of the simulated population approaches a stable age distribution at the carrying capacity, which was similar to that generated by the Leslie matrix. The logistic equation has been applied to study many phenomena, and the matrix model can be applied to these same processes. For example, random variation can be added to life history parameters, and population abundances generated with random effects on fecundity show both the affect of annual variation in fecundity and a longer-term pattern resulting from the age structure.     

A density-dependent Leslie matrix model

A density-dependent Leslie matrix model introduced in 1948 by Leslie is mathematically analyzed. It is shown that the behavior is similar to that of the constant Leslie matrix. In the primitive case, the density-dependent Leslie matrix model has an asymptotic distribution corresponding to the logistic equation. However, in the imprimitive case, the asymptotic distribution is periodic, with period depending on the imprimitivity index.     

用Leslie矩阵法预测人口——四川省彭县清平公社1978年人口资料分析

人口分析是种群生态学研究的主要对象之一。Leslie矩阵则是Leslie(1945)提出的分析动物种群数量变动的一种数学模型。现在它的应用已经推广到资源管理、生态系统的分析等许多方面(Jeffers,1978;Pallard,(1973),Leslie曾利用1960年澳大利亚女性人口资料,从理论上讨论了他自己提出的随机模型的性质。 本文的打算是:借Leslie矩阵法,用四川省彭县清平公社1978年(指从1978年7月1日到1979年6月30日的人口统计年)的人口调查资料作典型,来预测川西平原农村人口的发展趋势和稳定人口分析,从而对四川省在控制人口中采取的人工流产、引产等计划生育补救措施作初步评价。     

高原鼠兔种群生产量生态学的研究Ⅲ.高原鼠兔生命率非密度制约与密度制约的种群动态趋势

本文以高原鼠兔(Ochotona curzoniae)自然种群生命表的统计参数为基础,根据非密度制约Leslie模型及具有密度制约反馈的标准Leslie修正模型,分别预测了该种群在1982-2001年间的发展趋势。在菲密度制约条件下,该种群呈指数增长。在密度制约存在肘,种群增长趋于平衡状态,且存滔率密度制约较繁殖率密度制约对种群的作用更大。存活率密度制约与非密度制约的年龄结构均为Leslie分布,繁殖率密度制约作用的种群稳定年龄分布更平均,其平衡状态的种群大小则由模型的参数决定。     

Asymptotic properties of infinite Leslie matrices

The stable population theory is classically applicable to populations in which there is a maximum age after which individuals die. Demetrius [1972. On an infinite population matrix. Math. Biosci. 13, 133-137] extended this theory to infinite Leslie matrices, in which the longevity of individuals is potentially infinite. However, Demetrius had to assume that the survival probability per time step tends to 0 with age. We generalise here the conditions of application of the stable population theory to infinite Leslie matrix models and apply these results to two examples, including or not senescence.     

Bias in estimating the Malthusian parameter for Leslie matrices

For Leslie matrices of order 3 × 3 or larger, conditions for concavity or convexity of the Malthusian parameter in each of the entries in the matrix are given. Both cases are possible so it follows that the expected population growth rate computed from a Leslie matrix whose entries are random variables can be either smaller or larger than the growth rate computed from the expected value of the matrix. Boyce [(1977) Theor. Pop. Biol.12] showed that in the 2 × 2 case this bias is always positive; we give a numerical example illustrating the magnitude of the bias in this case, and compare it with the sampling error of the parameter for the same example.     

陕西不同地区栓皮栎种群年龄结构动态模型的研究

栓皮栎种群动态模型能阐明其自然种群动态的规律,揭示种群的内在机制和对种群行为进行预测。矩阵模型是一种i状态分布方法,依靠矩阵形式来处理种群的特征分布,可以模拟和预测种群中各个年龄组的数量动态和年龄结构的变化,它能够从目前已知的年龄结构及种群的生存率和生育率,来推测种群的未来年龄结构。在研究栓皮栎种群动态时,也借用该模型对栓皮栎种群各年龄组的结构和数量动态作出预测。Leslie矩阵模型就是该模型中的一种,利用该模型的理论和方法,对栓皮栎种群的自然变化过程进行了模拟和预测。结果发现,从分布中心到分布边缘,随着生境的差异,栓皮栎种群产生幼苗的年龄级、内禀增长率、生育力和幼苗的数量都发生变化,预测结果与实际反映的情况基本一致,表明Leslie矩阵模型是一种较为理想的反映种群动态的模型。模型表达形式简单,参数生态学意义确切,应用精度高,从而达到准确预测栓皮栎种群动态的目的。     

Stable population analysis in periodic environments

A discrete time model is developed for periodic survivorship and maternal frequency rates. The Leslie matrix is subdivided by an additional variable representing time of birth (season of birth in the example presented) to accommodate both age-specific and time-specific variations in vital rates. Thus, in contrast to the standard time-invariant model, significant periodic alterations in age-specific birth and death rates are explicitly accounted for and may realistically include observed recurrent changes, such as zero or reduced birth rates during unfavorable seasons, etc. Conditions for stability of the extended projection matrix are developed and are shown to be analogous to those of the Leslie model. The periodic model is applicable to populations with overlapping generations in seasonal environments.     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Simultaneous determination of fluctuating age structure and mortality from field data

The use of a Leslie matrix for analysis of a population normally implies that the age structure of the population is known. However, this restriction can be overcome if the population can be partitioned into recognisably different stages, and some information on stage duration and fecundity is available, in which case the age structure may be determined by the analysis itself. As an example of this approach we consider the estimation of the mortality rate applying to a population from a sequence of observed stage frequency vectors. The technique does not require that the population has attained a stable age structure nor that distinct cohorts can be recognised.     

Raising Leslie matrices to powers: a method and applications to demography

An illustrative method, labelled Strip and Mask, to raise a Leslie matrix to powers is introduced. Starting from a recent article in this journal, the Strip and Mask method is utilized to determine the primitivity pattern of a Leslie matrix, and to discuss some properties of the corresponding population model.     

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.