种间竞争的一个新的数学模型—对经典的Lotka—Vol—terra竞争方程的扩充

本文根据营养动力学理论,建立了一类种间竞争的新的数学模型:它是单种群增长的Cui-Lawson模型,在种间竞争上的推广。新的种间竞争模型克服了经典的种间竞争的Lotka-Volteira方程的局限与不足,具有更广泛和复杂的行为,并在特殊条件下以Lotka-Volterra竞争方程为其特例。因此,新的种间竞争的数学模型是更一般的解释性模型,是对经典的Lotka-Voterra竞争方程的扩充。     

Competitive Exclusion Principle in Ecology and Absolute Asymmetric Synthesis in Chemistry

The key concepts underlying the Frank model (1953) for spontaneous asymmetric synthesis in chemistry are traced back to the pioneering works of Volterra (1926) and Lotka (1932) on biological species competition. The Lotka‐Volterra (L‐V) two‐species exclusive competition model reduces to the Frank model for the special case of distinguishable but degenerate species (i.e., the enantiomers). The important ecological principle of competitive exclusion, originally derived from the L‐V two‐competitors model, is a consequence of sufficiently antagonistic interactions between the species competing for limited common resources, or mutual inhibition, as the term is known in the chemical literature on absolute asymmetric synthesis. The L‐V and Frank models are described by the same general differential equations, nevertheless a crucial thermodynamic distinction between these models is necessary to correlate ecological selection and chemical selectivity arising from 1) the absence of reversibility in biological transformations, in marked contrast to chemical reactions, and 2) the constraints in chemical scenarios on the reaction rate constants required to fulfill the principle of micro‐reversibility. Chirality 27:722–727, 2015. © 2015 Wiley Periodicals, Inc.     

Demographic Models for Subdivided Populations: The Renewal Equation Approach

Le Bras (Theor. Popul. Biol.2, 100–121, 1971) and Rogers (Demography11, 473–481, 1974), in two neglected papers, have generalized to the multisite case the Euler–Lotka renewal equation and demographic characteristics such as age structure and reproductive value. The purpose of this paper is twofold: first, to restate the multisite renewal equation in the matrix context; second, to derive results on age structure, net reproduction rate, generation time, and sensitivities, as generalizations of the one site case. The potential of this approach for population biology is illustrated using a model of a black-headed gullLarus ridibunduspopulation.     

长苞铁杉群落种间竞争的研究

以长苞铁杉及其伴生树种在正常纯林和伴生树种为优势种组成的阔叶林中的优势度为环境容纳量,用重要值百分数求取种间竞争系数,采用Ｌotka-Volterra竞争方程研究长苞铁杉林的共优种群（长苞铁杉与其伴生树种）种间竞争关系,结果表明,平衡时,长苞铁杉及其伴生树上对优势度分别为７４．２２％和２５．７８％,说明长苞铁杉种群支配整个群落。     

Competitive exclusion in a two-species chemotaxis model

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka–Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.     

Calculating the intrinsic growth rate: comparison of definition and model

It was shown that well known equation r = ln[N(t2)/N(t1)]/(t2 - t1) is the definition of the average value of intrinsic growth rate of population r within any given interval of time t2-t1 and changing arbitrarity its numbers N(t). The common opinion considering the equation as suitable only for exponentially growing population was found to be incorrect. The fundamentally different approach is based on the calculation of r within the framework of demographic model, realized as Euler - Lotka equation or population projection matrices. However this model requires simultaneous realization of several assumptions improbable for natural populations: exponential change in population size, stable age structure and maintaining constant age-dependent birth and death rates. The calculation of r by definition requires the data on the dynamics of population numbers, whereas calculation on the basis of the model requires the demographic tables of birth and death rate, but not the population numbers. With the example of American ginseng it was shown that evalution of r by definition and model approaches could produce opposite results.     

Inhomogeneous stationary states in reaction-diffusion systems.

Inhomogeneous stationary solutions for reaction—diffusion equations are of interest with regard to the chemical basis of morphogenesis. Numerical means of determining such solutions and of testing their stability to small perturbations are examined, and applied to shape formation in three systems: the “Brusselator” and “Lotka—Volterra” mechanisms, and a simple model for slimemould aggregation.     

Neutral stability in many-species food webs

The predator-prey systems of Lotka and Volterra are modified to include efficiencies e_ij of converting prey biomass to predator biomass. For three or more species, it is shown that neutral stability results if e_ik = e_ije_jk for all transfers. When the number of species is odd, consistency of the differential equation system for changes in logarithms of biomass is also required.     

Inhomogeneous steady state distributions of species in predator--prey systems. A specific example

We present and study a specific example of emergence of an inhomogeneous steady state distribution in a non- Lotka—Volterra predator-prey model. With the aid of a linear and a non-linear bifurcation analysis, the dependence of this distribution on species mobilities as well as other important model parameters is discussed. It is seen that the behaviour of the resulting colonies of species seems to correctly mimic some of the features of real predator-prey systems.     

Using Daphnia physiology to drive food web dynamics: A theoretical revisit of Lotka-Volterra models

The Lotka–Volterra model is the most commonly used framework to describe the dynamics of ecological systems in which two species interact, one as a predator and the other as prey. Theoretical ecologists have since built on variants of these equations, frequently applying them to model the dynamics of algal-herbivore interactions in aquatic systems. In this study, we augment a Lotka–Volterra system by introducing a bioenergetically-explicit, ecophysiological model to examine how variations in resource allocation affect zooplankton growth and subsequently phytoplankton dynamics. Ingested material within a zooplankter's gut is separated into distinct internal congener pools that are used to support physiological processes occurring in a hierarchical direction: neurological functions, energetics, osmoregulatory maintenance, waste management, and finally growth. Consistent with the predictions of the “stoichiometric knife edge” theory, our analysis suggests that a balanced algal congener composition is required to optimize zooplankton internal congener saturations, resulting in a maximal allocation of energy to growth. In examining the advantages rendered by different strategies of minimum and optimum somatic quotas when experiencing phosphorus-enrichment conditions, we show that herbivores with narrow homeostatic bounds and animals with low minimum quotas (or depletion specialists) achieve optimal performance first. Our analysis also predicts patterns of multiple stable equilibria in which the same environmental conditions can be characterized by dramatically different prey-to-predator ratios. Importantly, abrupt shifts from one state to another can be induced not only by short-term variations in food abundance but also by variations in the nutritional quality of the prey. Our predictions have profound implications for connecting microscopic processes with macroscopic patterns and offer new insights into the multitude of factors that modulate food web dynamics.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

Autocatalysis in a stoichiometric model of energy metabolism]

A simple kinetic model of the energy metabolism with autocatalytic stoichiometric structure is analysed. The model includes the participation of ATP (a product of energy metabolism) in the activation of oxidation substrate. It is shown that the energy metabolism displays multiple steady states and autooscillations in the absense of all kinds of non-stoichiometric (allosteric, isosteric and cooperative) interactions. The results of the analysis are represented in the form of a parametric portrait. The domains of the parameter values of the oxidation substrate source are separated, which correspond to qualitatively different patterns of the model dynamic behavior. Under specified conditions the model is reduced to various versions of the Lotka model.     

The need for alternative plant species interaction models

对植物物种相互作用备选模型的需求 近年来，经典的Lotka-Volterra模型在分析和解释植物物种间竞争性相互作用方面的局限性愈发凸显。目前已经明确的3个问题是：(1)缺乏频率依赖性，其对于物种的长期共存至关重要；(2)需要考虑影响个体表现的未测量(通常不可测量)的变量(例如，土壤养分或病原体的空间变化)；以及(3)需要将测量误差与生物变异度分开。本文改进了经典的Lotka-Volterra竞争模型以期突破模型的局限。同时，我们将8个备选模型与丹麦草本植物群落中羊茅(Festuca ovina)和细弱剪股颖 (Agrostis capillaris) 3年以上的盖度数据拟合，应用贝叶斯建模框架来确定模型改进是否提高了模型的性能，并提高了其预测群落动态并因此检验假设的能力。研究结果表明，纳入频率依赖性和测量误差极大地改进了模型性能，但将可能的未测量变量考虑在内却未能改进其性能。我们的研究结果强调了在植物群落动态的定量研究中比较备选模型的重要性。只有考虑可能的备选模型，我们才能确定驱动群落构建和变化的因子，并提高我们预测植物群落行为的能力。     

General continuum analysis of transport through pores. II. Nonuniform pores.

A general continuum derivation of the nonelectrolyte (Js) and volume (Jv) flux through a pore whose cross section is a function of axial position (nonuniform) is given. In general, the flux equations cannot be reduced to the same form as for a uniform pore and it is not possible to characterize the pore kinetics by three constants as in the uniform pore case. However, it is shown that under certain conditions, the nonuniform pore equations can be approximated by the uniform pore form and can be characterized by three constants (omega, sigma, Lp). The only condition needed to reduce the Jv equation to the uniform form is that the solution be dilute. The deviation of the Js equation from the uniform form is characterized by an asymmetrical function of Jv whose maximum value is estimated. It is shown that the maximum posible fractional deviation of the Js equation from the uniform form is given by the parameter: 0:5sigmaJv/omegaRT. Since this parameter is less then 0.15 for most membrane studies, the nonuniform Js equation can usually be approximated by the uniform pore form. The general results are illustrated by explicit calculations on several models of nonuniform pores. It is shown, for example, that the "equivalent pore radius" defined in the usual way is a function of the experimental parameter that is measured and is not unique.     

Focus

A solid background in population ecology is valuable for anyone concerned with natural resources management. However, without the opportunity of field experience, it may be difficult for students to become familiar with some of the principles of population biology. Microcomputers and programmable pocket calculators allow the construction of simple simulation models, making it possible to carry out some ‘numerical experiments’ in the classroom which may play the same role in the learning process as in vivo experimentation

This work introduces a mathematical model based on the Lotka—Volterra equations and a computer program developed for a programmable calculator. They can be used to assess the influence of environmental heterogeneity and disturbances on the results of competitive relationships between two plant populations. As an example, the interaction between two tussock grass species of the Patagonian steppe, Stipa speciosa Trin. et Rupr. and Festuca pallescens (St Yves) Parodi (the second being less xerophytic than the first one), is analysed.     

A rigorous model study of the adaptive dynamics of Mendelian diploids

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka–Volterra-like dynamics and single locus genetics (as in Kimura’s in Proc Natl Acad Sci USA 54: 731–736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the ‘long’ time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M ₁ topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as ‘canonical equation’.     

Behavior of pesticides (1): efficiency evaluation of pesticides.

The correlation equation for the concentration-mortality curve of a specific pesticide and a control pesticide was used to adapt the effectiveness data from residual activity tests to the corresponding chemical application concentration, to calculate the constant with the decreasing speed equation, and to calculate the application dose necessary to maintain an equal residual effect period by the same equation. The calculated value is the amount of pesticide needed to perform with the same efficiency as the control pesticide (effectiveness and period of effectiveness), which is vitally important to effectively select an appropriate pesticide.     

Community diversity and total abundance: Quantitative predictions from competition niche theory

Lotka–Volterra niche competition theory (LVNCT) is based on Lotka–Volterra competition equations with competition coefficients between pairs of species determined by the intensity of their niche overlap though the MacArthur–Levins niche overlap formula. Here I study analytically and numerically the predictions of LVNCT concerning total abundance and biodiversity, measured by the Shannon equitability index. Firstly, a set of simplifying assumptions that render the LVNCT amenable of analytical treatment are considered. In particular I derive an approximated formula for the total abundance, as the inverse of the mean value of the interspecific competition coefficients, which works pretty well both for the transient and steady regime and for a wide range of the typical niche width σ. Secondly, I analyze, by means of simulations, the effect of relaxing the above simplifying assumptions when considering more realistic conditions. It turns out that the approximated formula for the total abundance is quite robust and its potential implications for management are discussed. I also analyze the predicted relationship between community productivity and diversity.     

On the reproductive value and the spectrum of a population projection matrix with implications for dynamic population models

The eigenvalues of a population projection matrix-except for the Lotka coefficient-are uniquely determined by the reproductive values and the survival. This relation (proposed earlier, but not really well known in western literature) follows from another useful relation between fertility, reproductive values, survival, and Lotka’s coefficient. These results are applied to provide demographic interpretations to the intrinsically dynamic and metastable population models by Schoen and co-workers.