广义LESLIE矩阵模型的渐近分析

考虑昆虫种群的不等时间间隔的虫态历期和不同个体的历期差异,沈嘉骥等提出了广义Leslie矩阵模型。这是把Leslie矩阵模型推广到昆虫种群动态研究中一个很有意义的新进展。本文拟对该模型从数学理论上进行较深入的讨论。我们将看到,它是Leslie模型数学理论的一个自然推广。先将模型转述如下,我们用年表时间单位。设考虑的昆虫种群有m个虫态。N_i(k)=(n。(k),…,nτdi(k))(τ=1,…,m;k=1,2,…)。表示第七年的第i个虫态的年龄结构,d_i是第i虫态的最长历期。第k年的种群向量为n~T(k)=(n_1~T(k),…,n_m~T(k)),它的维数     

广义LESLIE矩阵模型的稳定年龄结构及其性质

〔1〕提出了一个描述生物种群动态的矩阵模型——广义Leslie矩阵模型。这个模型是Leslie矩阵模型的一个推广,它考虑了昆虫发育过程中个体在虫态发育历期上的差异。因此与普通Leslie矩阵模型相比,该模型更适于描述昆虫种群的动态。〔2〕对广义Leslie矩阵模型的渐近性质做了分析,给出了相应的稳定年龄结构的表达式,同时从马氏链的观点出发,给出了种群年龄结构渐近稳定或周期变化的充要条件。本文是〔2〕的继续,将给     

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。     

具有时间依赖转移率的昆虫种群的稳定性理论

研究Leslie种群的稳定性一直是种群动态研究中的重要课题。近几年,这项工作已得到比较完满的解决。考虑到昆虫个体在发育历期上的差异,〔1〕对Leslie模型进行了推广,提出了广义Leslie矩阵模型。〔2〕在转移率为常数的情况下,讨论了有关广义Leslie种群的渐近稳定性等方面的问题。本文将研究具有时间依赖转移率的广义Leslie种群的稳定性,给出一个具有时间依赖转移率的昆虫种群渐近于一个稳定的年龄分布的充分条件。     

濒危植物长柄双花木自然种群数量动态

 运用种群生命表、生殖力表、Leslie矩阵模型和时间序列预测分析方法,研究了濒危植物长柄双花木(Disanthus cercidifolius var. longipes)种群的动态变化过程,揭示了长柄双花木各龄级植株的数量动态规律。结果表明：长柄双花木为缓慢负增长型种群;种群的净增殖率、内禀增长率以及周限增长率都较低,世代平均周期较长;Leslie矩阵模型和时间序列预测分析均表明在未来50年内长柄双花木各年龄级种群数量会出现波动性的消长,但其种群总数将逐步下降。导致种群下降的主要原因可能是人为砍伐及由此造成的生境破碎化等。     

由大熊猫的年龄结构看其种群发展趋势

四川卧龙自然保护区“五一棚”的大熊猫种群年龄结构已有报道,现据此数据通过Leslie矩阵预测该种群的发展趋势。按3年一个年龄组调整了1978年的数据,估计了各年龄组内1只大熊描3年内出生的幼仔数。并以魏辅文同学的数据,计算出各年龄组的存活率。 预测到2011年,发现该种群增长缓慢。在其种群年龄结构发展到正常状态,即呈金字塔形后(1978年2-5岁组数量太低,结构不正常),年增长率也不过1.66%或1.64%。 “五一棚”是大熊猫最好的栖息地, 保护措施又最得力,增长尚如此缓慢,其他地区恐还达不到这个数字。因此,动物保护事业必须加强,保护措施必须严格执行。     

人口增长的数学模型探讨

本文根据人类种群与生物种群的共性与特点,给出一种人口增长的数学模型,并从生态学意义上作一些探讨。一、人口增长的确定性模型在人类种群的动态变化过程中,由于人类世代之间有重叠,因此宏观上可以认为种群数量是连续变化的,可以用一个实值连续函数x=x(t)表示t时刻人类种群的大小。人口运动的基本动态模型可表示为:     

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。     

滇牡丹自然种群数量动态

运用种群静态生命表、存活曲线、生殖力表和Leslie矩阵模型研究了中国西南特有濒危植物滇牡丹(Paeonia delavayi)种群数量动态过程。静态生命表和种群存活曲线表明: 滇牡丹在株龄3-6 a之间经历了较强的环境筛, 其单株生理寿命为15 a左右, 平均周期为8 a, 种群的净增殖率(R₀ = 0.985 7)、内禀增长率(r_m = -0.001 7)和周限增长率(λ = 0.998 3)表明其为衰退型种群; 滇牡丹种群存活表现为台阶型曲线(B1型), 分别在6 a和12 a阶段种群消亡率(K_x)较高。Leslie矩阵模型的模拟结果表明, 在未来30 a内种群数量呈现出下降趋势, 大约下降了50%, 其种群数量靠自身根系的萌蘖和种子繁殖共同维持。野生滇牡丹种群数量下降与其自身繁殖特性有关, 但主要原因是人为采挖和生态环境的破坏。     

高寒草甸生态系统牲畜种群结构和动态

本文用Leslie矩阵模型研究了高寒草甸生态系统牲畜种群结构及动态。模型考虑了更加精确的年龄组转移关系,出栏率是种群波动的主要因子。目前,牲畜种群结构不合理,种群数量不能保持平衡。     

猕猴(Macaca mulatta)生命表研究

本文研究了猕猴雌性生命表。南湾猕猴各年度全群平均存活率高于93％,特定年龄组逐年存活率直到13岁仍达0.81以上,种群内禀增长率为0.1327,年周增长率为1.1419。用莱斯利矩阵对种群发展作预测,将预测值与1985-1987年的实测值作卡方适应性检验,两者无显著性差异。进一步分析发现该方法预测南湾猕猴种群动态,在7年内是有效的。     

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.     

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.     

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.     

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.     

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.     

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.     

在年龄结构化动物种群的分析过程中常见的几种错误

分析动物各年龄组的种群数据有助于推断影响动物种群发展趋势的诸因素,而且便于设计出种群层次。然而,在研究近期文献资料时,我们发现存在三种错误。这些错误均可导致非正确的解释。即:1)在生命表分析中,将Sx与lx等同(Sx由各死亡年龄组推导而出,lx为存活率);2)在综合跨年龄组的成雌生育率时,不计算该年龄组的成雌死亡数;3)在用Leslie矩阵计算周限增长率时,以mx取代Fx。我们将探讨上述错误的含义及后果,并提出解决方法。