一个新的种群增长数学模型——对经典的logistic方程和指数方程的扩充

一、引言 最简单的种群数学模型是众所周知的指数方程: dX/dt=μ_eX (la)为了今后讨论方便,现写出其积分形式: 1+n X_2/X_1=μ_e(t_2-t_1) (ab)此处,t,t_1,t_2为时间,X,X_1,X_2分别为相应时间的种群密度;μe为增殖速度参数,单位为(时间~(-1)) 但是指数增殖只能维持一个有限时段。随着群体增大,资源要被消耗,增殖受到限制。考虑到此,另一个著名的logistic方程常被采用:     

柳杉—杉木混交林种间竞争的研究

以混交树种在正常林分中的蓄积量为其环境容纳量,采用蓄积量百分数求混交林种间竞争系数,用Lotka-Volterra竞争方程探讨柳杉－杉木混交林种间竞争关系。结果表明,平衡时,柳杉、杉木相对蓄积量分别为76．21％和32．79％, 说明两树种能协调生长。     

入侵植物藿香蓟表型可塑性对种间竞争的响应

本文以恶性入侵植物藿香蓟（Ageratum conyzoides L.）与其伴生种紫苏（Perilla frutescens (L.) Britt.）为研究对象,分析不同种间竞争距离下藿香蓟功能性状的表型可塑性。结果显示：（1）较高的藿香蓟初始高度可显著减小其后续的相对株高生长（回归系数β=-0.137）,相对株高生长与其初始高度呈显著负相关,藿香蓟的株高生长速度显著高于紫苏;（2）竞争等级对藿香蓟收获时的地上生物量、比茎长和花蕾强度有显著影响（P <0.05）,在竞争距离为5 cm时,藿香蓟比茎长和花蕾强度显著高于其他组别;（3）竞争的不对称性仅影响藿香蓟的比茎长,初始高度高于藿香蓟的紫苏株数越多,藿香蓟的比茎长就越长（β=3.374）;(4)除高生长,藿香蓟其余7个性状均表现出较高的表型可塑性,变异系数CV值在0.17～0.55。藿香蓟综合表型可塑性在不同竞争距离下存在显著差异,在紫苏-藿香蓟距离为2.5 cm时,藿香蓟表型可塑性指数低,竞争距离为10或20 cm时,可塑性指数显著提高。综合来看,面对紫苏的近距离竞争（2.5 cm）,藿香蓟的生长和繁殖普遍受到抑制,表现为较低...     

温度对两种蓝藻种间竞争的影响

通过室内实验研究了不同温度条件下主要水华藻类——铜绿微囊藻(Microcystis aeruginisa)和巨颤藻(Oscillatoria princeps)的生长和种间竞争。结果表明:无论在纯培养体系还是混合培养体系中,微囊藻在25℃下生长最好,颤藻在30℃下生长最好;温度对藻类的种间竞争抑制参数能够产生明显影响,在20℃、25℃、30℃3个温度下,颤藻对微囊藻的竞争抑制参数随温度的升高而升高,30℃时颤藻对微囊藻的竞争抑制参数分别是20℃和25℃时的1.42倍和1.13倍,微囊藻对颤藻的竞争抑制参数则表现为25℃>30℃>20℃,25℃时微囊藻对颤藻的竞争抑制参数分别是20℃和30℃时的1.54倍和1.21倍,在20℃、25℃、30℃3个温度下,微囊藻对颤藻的竞争抑制参数均大于颤藻对微囊藻的竞争抑制参数,说明在各试验温度下,微囊藻对颤藻的抑制能力均大于颤藻对微囊藻的抑制能力;根据Lotka-Volterra竞争模型中的两物种竞争结果可初步判断,20℃和25℃温度下,微囊藻在竞争中取胜,30℃下,微囊藻和颤藻可以实现不稳定共存。     

克隆植物对种间竞争的适应策略

克隆植物种群因其寿命的持久性、空间上的可移动性和繁殖方式的多样化等特征与非克隆植物有很大区别, 在自然生态系统中占有重要地位, 甚至成为优势种或者建群种。该文通过归纳有关克隆植物的种间竞争适应策略研究案例, 阐述了克隆植物的竞争能力差异和影响竞争力的因素; 论述了克隆植物在构件形态、克隆构型、繁殖对策等方面对种间竞争的响应, 以及生理整合作用与种间竞争的关系; 分析了导致某些同类研究的结论不一致的原因, 认为实验对象差异、实验设计、生境条件与克隆植物形态及生理上的时空动态变化等都可能影响实验结果; 提出了全球变化背景下的克隆植物种间竞争及其分子生态学机制等可能是今后需要重点关注的问题。     

五种典型藻类对C，N，P吸收动力学分析及种间竞争模拟

目的:获得五种典型藻类(甲藻属微小亚历山大藻(Alexandrium minutum)和锥状斯氏藻(Scrippsiella trochoidea),赤潮硅藻中肋骨条藻(Skeletonema costatum)以及绿藻属杜氏藻(Dunaliella salina)和青岛大扁藻(Platymonas helgolandica tsingtaoensis))对C,N,P营养的吸收动力学参数,并利用经典藻类种间竞争模型,构建一个藻类混合共存的生态平衡体系,揭示藻类种间竞争规律,为赤潮爆发机制和预测的研究提供一个新思路。方法:监测批次培养过程中藻体的生长规律以及培养基中C,N,P营养的消耗,计算出藻类营养吸收动力学参数,将参数代入Huisman-Weissing竞争模型,模拟藻类种间竞争。结果:(1)在单独培养条件下,杜氏藻具有最高的比生长率(0.834 d~(-1))和最大细胞浓度(3.4×10~6 cells/mL),锥状斯氏藻和微小亚历山大藻的比生长率μ和最大细胞浓度与其它三种藻相比均明显偏低,p0.01;(2)随着环境总碳浓度从5 mM提高到20 mM,五种藻的比生长率和最大细胞浓度均显著上升,其中杜氏藻和青岛大扁藻对C浓度改变的响应更加敏感;(3)杜氏藻和中肋骨条藻理论最大比生长率(μmax)明显高于其它三种藻类,锥状斯氏藻和微小亚历山大藻对C,N,P营养盐的需求量相比于其它三种藻明显偏高,p0.01;(4)藻类共生平衡系统中,N营养添加有利于杜氏藻和中肋骨条藻发挥更好的种间竞争优势,P营养添加有利于微小亚历山大藻和锥状斯氏藻发挥种间竞争优势;结论:不同环境条件下,五种藻类最大比生长速率μmax和营养吸收半饱和常数Ks直接影响它们的种间竞争能力,基于藻类动力学参数的种间竞争模型为赤潮爆发机制和预测的研究提供一个新思路。     

种间竞争对香蒲与芦苇生长的影响

为了探究种间竞争对香蒲（Typha domingensis）与芦苇（Phragmites australis）生长的影响,利用根系分隔盆栽试验研究了3种不同分隔方式条件下香蒲与芦苇的种间竞争特性,包括塑料膜分隔（根系完全分隔,无相互作用,无物质交换）、尼龙网分隔（根系部分分隔,无相互作用,有物质交换）和不分隔（根系完全相互作用,有物质交换）,分析了香蒲与芦苇根系形态和地上部生长的变化,探究香蒲与芦苇种间竞争的差异。结果发现（1）在尼龙网分隔和不分隔处理中芦苇具有明显的劣势。与塑料膜分隔处理相比,芦苇的总生物量、植株密度在尼龙网分隔和不分隔处理中分别减少了39.14%、49.41%和82.08%、79.22%,总根长、总根表面积、总根体积分别减少了40.53%、44.84%、62.52%和85.7%、82.45%、89.67%,且均具有极显著差异（P<0.01）;根系分隔方式也影响芦苇的株高、茎粗和叶片数,表现为不分隔 > 塑料膜分隔 > 尼龙网分隔。（2）与塑料膜分隔处理相比,香蒲总生物量在尼龙网分隔和不分隔中虽有增加,但差异不显著,植株密度和株高在尼龙网分隔和不分隔处理中都有增加且具有显著差异（P<0.05）,其总根长、总根表面积、总根体积在尼龙网分隔中分别增加了57.93%、26.5%、8.04%,但在不分隔处理中分别减少了11.57%、14.92%、11.39%（P<0.05）,虽然根系的相互作用对香蒲根系的生长具有促进作用,但植物种间根系相互作用越强,对两者的生长越不利。（3）在不同的分隔方式中,芦苇与香蒲间也存在明显变化。在不分隔处理中,香蒲的生物量和植株密度是芦苇的1.7倍和6.74倍,与塑料膜分隔处理相比增加了6倍,表明芦苇与香蒲根系的完全相互作用,显著削减了芦苇的繁殖生长,增加了香蒲的根系分蘖。（4）通过种间相互作用强度（RII值）分析也表明,尼龙网分隔和不分隔处理下芦苇表现为抑制作用（RII值为负值）,香蒲表现为促进作用（RII值为正值）。香蒲与芦苇互作对芦苇具有抑制作用,说明种间相互作用是能改变植物的适应性和植物群落的繁殖,同时也表明植物根系不仅在吸收土壤中的水和养分中起着关键作用,在种间关系中也起着重要作用。因此利用种间竞争控制植物生长,可以为保护生物多样性和生态系统的功能提供有效的技术支撑。     

疏叶骆驼刺根系对土壤异质性和种间竞争的响应

近年来, 植物根系对土壤异质性的响应和植物根系之间的相互作用一直是研究的热点。过去的研究主要是针对一年生短命植物进行的, 而且多是在人工控制的温室条件下进行的。而对于多年生植物根系对养分异质性和竞争的综合作用研究很少。该文对塔里木盆地南缘多年生植物疏叶骆驼刺(Alhagi sparsifolia)根系生长对养分异质性和竞争条件的响应途径与适应策略进行了研究, 结果表明: (1)在无竞争的条件下, 疏叶骆驼刺根系优先向空间大的地方生长, 即使另一侧有养分斑块存在, 其根系也向着空间大的一侧生长; (2)在有竞争的条件下, 疏叶骆驼刺根系生长依然是优先占领空间大的一侧, 但是竞争者的存在抑制了疏叶骆驼刺的生长, 导致其枝叶生物量和根系生物量都明显减少(p < 0.01), 而养分斑块的存在促进了疏叶骆驼刺根系的生长; (3)疏叶骆驼刺根系的生长不仅需要养分, 也需要足够的空间, 空间比养分更重要; (4)有竞争者存在的时候, 两株植物的根系都先长向靠近竞争者一侧的空间, 即先占据“共有空间”。研究结果对理解植物根系觅食行为和植物对环境的适应策略有重要意义。     

两种赤眼蜂对小菜蛾卵的寄生能力和种间竞争

在实验室内研究了本地天敌拟澳洲赤眼蜂和外来天敌短管赤眼蜂在不同卵龄和不同空间条件下对小菜蛾卵的寄生和种间竞争.结果表明,在0～62 h的小菜蛾卵上,拟澳洲赤眼蜂和短管赤眼蜂单独接蜂时对小菜蛾卵的寄生率、子代羽化率、子代雌蜂百分率随卵龄下降,而后代发育死亡率则随卵龄增加.短管赤眼蜂对各龄小菜蛾卵的寄生率、子代羽化率、子代雌蜂百分率均较拟澳洲赤眼蜂高,而子代蜂的发育死亡率则较拟澳洲赤眼蜂低.两种蜂对小菜蛾卵龄的要求都不太严格,在小于48 h的小菜蛾卵上都能产卵寄生,且寄生率高于50%,但短管赤眼蜂寄生对小菜蛾卵龄的要求更宽松.在两种蜂混合接蜂时,后代中短管赤眼蜂的比例在各处理中都高于50%,且随寄主卵龄增加,当小菜蛾卵龄大于48 h后,短管赤眼蜂的比例高达100%,说明短管赤眼蜂对小菜蛾有更强的寄生能力和竞争能力.拟澳洲赤眼蜂和短管赤眼蜂单独接蜂时,在4～22cm^3空间范围内,各处理间对小菜蛾卵的寄生率没有显著差异,而短管赤眼蜂的寄生率(76.4%～86%)略低于拟澳洲赤眼蜂(88.7%～92.3%).当空间大于53 cm^3时,寄生率显著下降,在102cm^3空间时显著降低到50%.混合接蜂时各处理间寄生率差异不显著.两种接蜂方式对后代羽化率和雌蜂百分率没有太大影响,蜂的后代发育死亡率在两种蜂单独接蜂时随空间而增加.在4～102cm^3空间范围内,混合接蜂后代雄蜂中短管赤眼蜂从80%以上降低到20%以下,说明短管赤眼蜂的竞争能力随接蜂空间的加大而降低.     

干旱对红松和蒙古栎幼苗生长的影响及种间竞争

以红松(Pinus koraiensis Sieb.et Zucc.)和蒙古栎(Quercus mongolica Fisch.ex Ledeb.)幼苗为研究对象,针对东北地区未来可能的干暖气候条件,利用盆栽实验进行干旱条件下两种幼苗的生长与竞争研究,分析其光合生理参数和生物量变化。一个生长季的实验结果表明:(1)干旱致使红松和蒙古栎幼苗的光合速率均下降,蒙古栎降低幅度达到显著水平,红松下降不显著。干旱使蒙古栎的水分利用效率下降而红松却显著增加。干旱还导致蒙古栎幼苗的茎叶和侧根生物量以及全株叶面积下降,但却增加了红松的细根生物量和根冠比。(2)混栽栽植处理降低了蒙古栎幼苗的最大净光合速率和水分利用效率,进而减少了其茎和侧根生物量。而对于红松幼苗,混栽同样降低了其水分利用效率,但也增加了其比叶面积和全株叶面积,因而对幼苗生长影响较小。以上结果表明,蒙古栎对干旱反应更为敏感和迅速。种间竞争中红松幼苗比蒙古栎幼苗表现出更强的竞争能力。     

Asymptotic behaviour of solutions of the diffusive Lotka-Volterra equations

A spatially discrete version of the diffusive Lotka-Volterra equations is considered. Asymptotical spatial homogeneity of solutions of the equations with equilibrium, periodic or zero flux boundary conditions is proved without regard to crowding effects. The proof does not require the assumption of equal diffusion coefficients and the restrictions on the dimension of space and on the initial data, which are necessary in the spatially continuous model.     

Coexistence for systems governed by difference equations of Lotka-Volterra type

The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.     

A mathematical model on fractional Lotka-Volterra equations

While there are many mechanisms that may be involved in the regulation of body mass in humans and other animals, it is not so clear how much regulation is needed beyond the negative feedback effect of body mass itself. Here we model weight changes as a stochastic process, and show that it behaves approximately as an autoregressive process. Using published estimates of the energy cost of weight gain, the effect of weight on resting metabolic rate and the daily variation in intake and activity, we show that fluctuations in weight will be small. The effect of excess intake is also examined, and the assumptions and limitations of the model are discussed.     

基于改进的Lotka-Volterra种间竞争模型预测退化高寒草地人工恢复演替结果

结合青藏高原地区黑土滩型退化高寒草甸改建成人工草地后恢复过程中植物群落组分的数量特征变化,基于中心差分法、相对误差热图、ODE算法等研究方法进行计算机模拟,对Lotka-Volterra种间竞争模型进行非线性化改进,建立了适于高寒草地人工恢复演替过程中的竞争效应预测模型,与经典的Lotka-Volterra种间竞争模型进行对比验证,实证了改进模型的可解性和准确性。并分析了改进的Lotka-Volterra模型的系统动力学行为,预测人工恢复植物群落各组分的竞争结局,判断人工草地恢复演替状况。结果表明:(1)随着恢复年限的增加,栽培植物(垂穗披碱草或草地早熟禾)与原生植物在经历一个激烈的竞争阶段后逐渐趋于动态平衡,表明人工种植的垂穗披碱草、草地早熟禾等本土植物,可以促进青藏高原地区"黑土滩"型退化高寒草甸的有效恢复;(2)综合各分组的恢复演替阶段特征,可食牧草、栽培植物和顶极植物均从第7年开始竞争力呈现明显下降的趋势,因此建议在对人工草地的恢复管理中,7—10年间进行适度的人为干预,如施肥、灭鼠害以及适当去除有毒有害杂草等;(3)综合各类植物的竞争演替预测情况,均从第20年左右开始逐渐趋于动态平衡。因此,根据该模型分析,从生态恢复的角度来看,"黑土滩"型退化草地进行人工恢复至少需要20年以上,才能获得较为稳定的植物群落。     

Biodiversity in model ecosystems, I: coexistence conditions for competing species

This is the first of two papers where we discuss the limits imposed by competition to the biodiversity of species communities. In this first paper, we study the coexistence of competing species at the fixed point of population dynamic equations. For many simple models, this imposes a limit on the width of the productivity distribution, which is more severe the more diverse the ecosystem is (1994, Theor. Popul. Biol. 45, 227-276). Here we review and generalize this analysis, beyond the "mean-field"-like approximation of the competition matrix used in previous works, and extend it to structured food webs. In all cases analysed, we obtain qualitatively similar relations between biodiversity and competition: the narrower the productivity distribution is, the more species can stably coexist. We discuss how this result, considered together with environmental fluctuations, limits the maximal biodiversity that a trophic level can host.     

Advantageous indirect interactions in systems of competition

Ecological foodwebs display complex networks of species interactions. Here we discuss how two species, whether directly interacting or not, can crucially affect each other 'indirectly' through their mutual associations with intermediary species. A technique is presented for quantifying these "indirect effects", so that a simple measure emerges for the degree of overall harm or advantage that a particular species encounters from another in the context of a given foodweb. If the system is one of pure competition, the "direct" interaction between any two species is of course harmful to both. But when the method is applied to such systems it predicts that, if all interactions are allowed for including the indirect, in a large proportion of pair interactions at least one species will be deriving benefit from the other. Computer-generated samples confirm that this proportion is accurately predicted. The many pathways for indirect interaction can thus often reverse the direct connection, so that competing species achieve an interaction here called 'Advantageous in A Community Context' (ACC).     

A mathematical model of iron metabolism

A mathematical model of iron metabolism is presented. It comprises the following iron pools within the body: transferrin-bound iron in the plasma, iron in circulating red cells and their bone marrow precursors, iron in mucosal, parenchymal and reticuloendothelial cells. The control exerted by a hormone, called erythropoietin, on bone marrow utilization of iron for hemoglobin synthesis is taken into account. The model so obtained consists of a system of functional differential equations of retarded type. Most model parameters can be estimated from radiotracer experiments, others can be measured and numerical values can be assigned to the remaining ones making few reasonable assumptions according to the available physiological knowledge. Iron metabolism behavior under different therapeutical treatments was simulated. Model predictions were compared to experimental data collected in clinical routine.This work has been partially supported by C.N.R. (Italy) through grants N. 80.01227.07 and N. 81.00888.07     

A spatial model for the spread of invading organisms subject to competition

 A spatially explicit integrodifference equation model is studied for the spread of an invading organism against an established competitor. Provided the invader is initially confined to a bounded region, the invasion spreads asymptotically as a travelling wave whose speed depends on the strength of the competitive interaction and on the dispersal characteristics of the invader. Even an inferior, but established, competitor can significantly reduce the invasion speed. The invasion speed is also influenced by the exact shape of the dispersal kernel (especially the thickness of the tail) as well as the mean dispersal distance for each generation. Received 10 April 1996; received in revised form 21 August 1996     

A mathematical model of a crocodilian population using delay-differential equations

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.     

利用竞争指数评价水曲柳落叶松种内种间空间竞争关系

林木之间的竞争包括地上和地下两部分。竞争影响空间的确定对预测生长有重要的意义。本研究将竞争影响空间划分为两部分：距对象本最近的一圈,称为内圈,包括地上和地下竞争影响;除最近一圈以外称为外圈,主要为地下竞争影响。在此基础上将竞争指数分解为不同的竞争指数分量。并以水曲柳和落叶松纯林和混交林为例,定量地分析了竞争影响空间内地上竞争和地下竞争,种内和种间竞争影响大小。结果表明,纯林中两树体最近一圈内地上和