Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments.

In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka-Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka-Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized.     

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。     

A COEVOLUTIONARY ISOMORPHISM APPLIED TO LABORATORY STUDIES OF COMPETITION

Some empirical consequences of an isomorphism between the Lotka-Volterra competitive model and a coevolutionary competitive model are developed. In both the Lotka-Volterra and coevolutionary models, four competitive outcomes are possible: 1) species one wins, 2) species two wins, 3) indeterminate outcome, and 4) stable coexistence. These two models are isomorphic in the sense that the inequalities associated with a particular competitive outcome of the Lotka-Volterra model correspond in a one-to-one manner with similar inequalities associated with the same competitive outcome of the coevolutionary model. The inequalities of the Lotka-Volterra model involve the competition coefficients themselves, while the inequalities of the coevolutionary model involve the genetic variances and covariances of the competition coefficients. The isomorphism suggests some alternative interpretations of the results of classical laboratory studies of competition. The Lotka-Volterra (or ecological) hypotheses postulate that the competition coefficients are constant and that genetic considerations play no role in determining the competitive outcome. By contrast, the evolutionary hypotheses derived from the coevolutionary model postulate that the competition coefficients are variables and that the genetic variances and covariances of the competition coefficients determine the competitive outcome. The isomorphism is applied to competitive exclusion and coexistence, and to competitive indeterminacy in Tribolium. In particular, the evolutionary hypotheses isomorphic to the two classical explanations of competitive indeterminacy, the demographic stochasticity and genetic founder effect hypotheses, are constructed. The theory developed here and in a previous paper (Pease, 1984) provides one perspective on the relation among the Lotka-Volterra competition theory, quantitative genetics, competitive exclusion, the reversal of competitive dominance, coexistence, competitive indeterminacy in Tribolium, and experiments investigating the relation between genetic variability and the rate of evolution of fitness.     

随机时滞Lotka-Volterra模型数值解的收敛性

通常情况下,随机时滞Lotka-Volterra模型没有解析解,因而数值逼近方法是研究其性质的有效工具.本文根据Euler数值方法,利用鞅不等式和Ito公式讨论了一类随机时滞Lotka-Volterra模型数值解的收敛性,给出了数值解收敛于解析解的条件.最后通过数值算例对数值计算方法进行了验证.     

基于反馈控制离散Lotka-Volterra竞争模型的持续生存和稳定性

针对一类具有偏离自变量的离散Lotka-Volterra竞争模型,考虑到不可避免的外界扰动,通过引入反馈控制,基于一定的分析技巧得到该系统持久性与全局稳定性的充分条件.生态意义上表明:在外界扰动下,具有偏离自变量的离散Lotka-Volterra竞争模型仍能持续生存并保持全局稳定发展.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

非自治Lotka-Volterra型捕食扩散系统中有害时滞对该系统持久性的影响

研究了时滞对一类非自治Lotka-Volterra型捕食扩散系统的影响,该系统由n个斑块组成,食饵种群可以在斑块间迁移,而摘食者限制在某一个斑块不能扩散．我们假设密度制约项系数并不总是严格正的．通过运用比较定理及时滞泛函微分方程的基本原理,分两种情况表明了在一定条件下系统是一致持久的．两种情况的结果表明时滞的引入和变化即可能是“有害”,也可能是”无害”．进一步还说明了系统在一致持久性的条件下至少存在一个正周期解．这些结果是对已知的非自治Lotka-Volterra系统的一些结果的推广与改进．     

Mathematical models for continuous culture growth dynamics of mixed populations subsisting on a heterogeneous resource Base: I. Simple competition

The classical Monod model for bacterial growth in a chemostat, based on a Michaelis-Menten kinetic analog, is restated in terms of an approximate Lotka-Volterra formulation. The parameters of these two formulations are explicitly related; the new model is easier to work with, but yields the same results as the original. The model is then extended to the case where multiple alternate substrates may be growth limiting, using the corresponding kinetic analogs for multiple-substrate enzymes. Again, one is led to a Lotka-Volterra analog. In the multiple-substrate model, however, coexistence of multiple genotypes is possible, in contrast to the single-substrate model. The usual Lotka-Volterra conditions for existence and stability of pure or mixed equilibria may all be translated into corresponding statements about the parameters of the chemostat system. Possible extensions to deal with metabolic inhibition, cross-feeding, and predation are indicated.     

Analysis of seasonal fluctuations in the Lotka-Volterra model

A modification of the Lotka-Volterra model was proposed. The modification takes into account the factor of seasonal fluctuations in a "predator-prey" model. In this modification, interactions between species in summer are described by the Lotka-Volterra equations; in winter, individuals of both species extinct. This generalization makes the classic model unrough, which substantially extends the field of its application. The results of numerical simulation illustrate the statement formulated above.     

Varieties of mutualistic interaction in population models

A Lotka-Volterra model of mutalism indicates eight possible cases, of which two lead to survival of both populations, two indicate inevitable extinction, and four are indeterminate, the result depending on the initial population sizes. Conventional neighborhood stability analysis is a poor indicator of the biological result expected. Modification of the Lotka-Volterra model to give non-linear isoclines is necessary to obtain a minimum of biological realism; this modified model is illustrated with an analysis of a legume-Rhizobium mutualism.     

The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka-Volterra problem

This comment addresses critics on the claimed stability of solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem, proposed by Dubey al. (2010. A solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem using Boubaker polynomial expansion scheme. Journal of Theoretical Biology 264, 154-160). Critics are based on incompatibilities between the claimed asymptotic behavior and the presumed Malthusian growth of prey population in absence of predator.     

Qualitative permanence of Lotka–Volterra equations

In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.     

The effects of interference competition on stability,structure and invasion of a multi-species system

Summary An interference competition model for a many species system is presented, based on Lotka-Volterra equations in which some restrictions are imposed on the parameters. The competition coefficients of the Lotka-Volterra equations are assumed to be expressed as products of two factors: the intrinsic interference to other individuals and the defensive ability against such interference. All the equilibrium points of the model are obtained explicitly in terms of its parameters, and these equilibria are classified according to the concept of sector stability. Thus survival or extinction of species at a stable equilibrium point can be determined analytically.The result of the analysis is extended to the successional processes of a community. A criterion for invasion of a new species is obtained and it is also shown that there are some characteristic quantities which show directional changes as succession proceeds.     

Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment

The dynamics of single populations up to ecosystems, are often described by one or a set of non-linear ordinary differential equations. In this paper we review the use of bifurcation theory to analyse these non-linear dynamical systems. Bifurcation analysis gives regimes in the parameter space with quantitatively different asymptotic dynamic behaviour of the system. In small-scale systems the underlying models for the populations and their interaction are simple Lotka-Volterra models or more elaborated models with more biological detail. The latter ones are more difficult to analyse, especially when the number of populations is large. Therefore for large-scale systems the Lotka-Volterra equations are still popular despite the limited realism. Various approaches are discussed in which the different time-scale of ecological and evolutionary biological processes are considered together.     

具无限时滞的Lotka-Volterra 方程的持续性

本文研究了具无限时滞的Lotka-Volterra方程的持久性问题,给出了保证系统持久性的充分条件．     

通过线性状态反馈使四种群Lotka—Volterra模型永久持续生存

本文研究了四种群Lotka0-Volterra模型的永久持续生存问题,给出了通过线性状态反馈使四种群Lotka-Volterra模型永久持续生存的一些充分和必要条件。     

A solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem using Boubaker polynomial expansion scheme

In this study, an analytical method is introduced for the identification of predator-prey populations time-dependent evolution in a Lotka-Volterra predator-prey model which takes into account the concept of accelerated-predator-satiety.Oppositely to most of the predator-prey problem models, the actual model does not suppose that the predation is strictly proportional to the prey density. In reference to some recent experimental results and particularly to the conclusions of May (1973) about predators which are ‘never not hungry’, an accelerated satiety function is matched with the initial conventional equations. Solutions are plotted and compared to some relevant ones. The obtained trends are in good agreement with many standard Lotka-Volterra solutions except for the asymptotic behaviour.     

一个具有竞争关系的Lotka-Volterra模型全局结构分析

本文利用微分方程定性理论,对一类具有竞争关系的两种群Lotka-Volterra模型进行了系统全局结构分析,并给出了相应的结论.     

Are circadian oscillators structurally unstable?

The common belief is that all biological oscillations are of limit cycle type. It is shown in this article that the phase response curves simulated on a two-species Lotka-Volterra linear (i.e. non-limit cycle type) oscillator, do look similar to those obtained by experimental methods by different workers. The form of the phase response curves, the existence of singularities and the mirror-image symmetry of opposite perturbations are modelled on the Lotka-Volterra system. The study, which is strongly indicative of the possibility that the underlying oscillator (or oscillators) is (are) not structurally stable, also indicates the necessity of designing critical experiments, capable of distinguishing between limit cycle and non-limit cycle oscillators, since the single-pulse phase resetting does nothing to distinguish between them.