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.     

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.     

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.     

Backwrd population projection by a generalized inverse

The Leslie population projection matrix may be used to project forward in time the age distribution or age-sex distribution of a population. As it is a singular matrix, it does not have an inverse, and so it is not clear that there is a corresponding procedure for backward projection. In terms of the eigenvalues and eigenvectors of the Leslie matrix, certain generalized inverses are constructed that can sometimes be used advantageously for backward projection.     

Primitivity and convergence to stability

An exact expression for the index of primitivity g of a Leslie matrix is obtained, which applies also to time-varying matrices which share an incidence matrix. Elapsed time (not time intervals) to primitivity is shown to depend only weakly on the discretization scheme used. A lower bound for speed of convergence to the stable (fixed or time-dependent as appropriate) state is given which depends sensitively on g.     

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。     

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.     

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     

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.     

特定雌性动物群周期性地获得稳定收获量的数学模型

利用Ｌｅｓｌｉｅ矩阵,给出了特定雌性动物群周期性地获得稳定收获量的数学模型,并对模型适用于集约化饲养家畜的特殊情况进行了讨论。     

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.     

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.     

Harvesting under density-dependent mortality and fecundity

The problem of optimal harvesting in equilibrium is considered in a Leslie matrix model in which both mortality and fecundity in all age-classes may be density-dependent. The conclusion is that the optimal strategy is of the two-age-class type, in common with results obtained previously for simpler models.     

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.     

根茎型木本克隆植物准噶尔无叶豆的种群数量动态

 根据根茎型木本克隆植物的特征, 不以种群的分株数量代表种群大小, 而尝试以不同茎级的根茎长度代表种群大小, 运用种群静态生命表、存活曲线、生殖力表和Leslie矩阵模型, 研究了准噶尔无叶豆(Eremosparton songoricum)的两个种群——A种群(46°31.09′ N, 88°33.06′ E, 紧邻乌伦古湖)和B种群(46°28.07′ N, 88°33.07′ E, 位于沙漠腹地)的种群数量动态。结果表明: 种群存活表现为Deevey-I型。A种群在中龄阶段受到的人为干扰较大, 死亡率出现高峰, 种群的净增长率(R₀)、内禀增长率(r_m)和周限增长率(λ)较低, 表现为衰退型种群, Leslie矩阵模型的模拟结果表明, 15 a内种群呈现下降趋势; B种群所受到的压力主要是干旱贫瘠的荒漠环境所导致的系统压力, 种群的R₀、r_m和λ值适中, 表现为缓慢增长型种群, Leslie矩阵模型的模拟结果表明, 15 a内种群呈现先下降、再上升的趋势。此外, 研究结果验证了Leslie矩阵模型可以扩展应用到根茎型木本克隆植物这类特殊生活型植物的种群数量动态研究上。     

An extension of the Leslie matrix model to include a variable immature period

An extension to the Leslie matrix is presented in which the age of transformation from immature to adult has a log—normal distribution. The major effect of this is shown to be on the second largest eigenvalue. The ratio of the largest to the second largest eigenvalue |λ₁/λ₂|, which is an index of the rate of approach to the stable age distribution, is greater in the new model, even though the value of λ₁ is effectively the same. The differences in the models are most pronounced where the population is subjected to a harvesting regime.     

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.     

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.     

Mask mitigates MAPT- and FUS-induced degeneration by enhancing autophagy through lysosomal acidification

Accumulation of intracellular misfolded or damaged proteins is associated with both normal aging and late-onset degenerative diseases. Two cellular clearance mechanisms, the ubiquitin-proteasome system (UPS) and the macroautophagy/autophagy-lysosomal pathway, work in concert to degrade harmful protein aggregates and maintain protein homeostasis. Here we show that Mask, an Ankyrin-repeat and KH-domain containing protein, plays a key role in promoting autophagy flux and mitigating degeneration caused by protein aggregation or impaired UPS function. In Drosophila eye models of human tauopathy or amyotrophic lateral sclerosis diseases, loss of Mask function enhanced, while gain of Mask function mitigated, eye degenerations induced by eye-specific expression of human pathogenic MAPT/TAU or FUS proteins. The fly larval muscle, a more accessible tissue, was then used to study the underlying molecular mechanisms in vivo. We found that Mask modulates the global abundance of K48- and K63-ubiquitinated proteins by regulating autophagy-lysosome-mediated degradation, but not UPS function. Indeed, upregulation of Mask compensated the partial loss of UPS function. We further demonstrate that Mask promotes autophagic flux by enhancing lysosomal function, and that Mask is necessary and sufficient for promoting the expression levels of the proton-pumping vacuolar (V)-type ATPases in a TFEB-independent manner. Moreover, the beneficial effects conferred by Mask expression on the UPS dysfunction and neurodegenerative models depend on intact autophagy-lysosomal pathway. Our findings highlight the importance of lysosome acidification in cellular surveillance mechanisms and establish a model for exploring strategies to mitigate neurodegeneration by boosting lysosomal function.     

Properties of the Leslie population matrix

A simple similarity transformation of the Leslie matrix renders certain properties obvious. In particular the characteristic polynomial, characteristic vectors and principal vectors can be explicitly written out. Bounds for the dominant root are given.