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.     

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。     

Global stability in Lotka-Volterra systems with diffusion

Summary Sufficient conditions for global stability inn-species Lotka-Volterra systems with diffusion are derived. Both continuous environments with 0-flux boundary conditions and environments consisting of discrete patches are considered.     

一类竞争Lotka-Volterra系统的持续生存和灭绝

对一类非自治竞争Lotka-Volterra系统,其相互作用系数为非负常数,S.Ahmad和A.C.Lazer研究了该系统并得到结果在一定条件下,系统的第n个种群趋于灭绝而前n-1个种群强持续生存.本文我们讨论一类较一般的非自治竞争Lotka-Volterra系统,其相互作用系数为一类连续函数,得到同样结果.     

Lotka-Volterra自治竞争系统的共存与绝灭

研究了一类自治n种群Lotka-Volterra竞争系统．建立了一部分种群绝灭,另一部分种群持续生存新的判别准则,推广了已知相关结果．     

一类离散Lotka-Volterra系统的灭绝和稳定性（英文）

考虑了一类受有毒物质影响的两物种非自治离散Lotka-Volterra系统.在平均增长率意义下,得到了其中一种物种灭绝的充分条件,证明了另外一种物种的稳定性.     

The adaptive dynamics of Lotka-Volterra systems with trade-offs

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.     

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.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

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.     

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。     

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.     

种群Lotka-Volterra捕食系统多个极限环的存在性

通过降维把高维系统平衡点的稳定必及根限环的构造用低维系统来判定和实现,给出了一个三种群Lotka-Volterra捕食系统具有两个小扰动极限环的例子。     

On the stabilizing effect of predators and competitors on ecological communities

Ecological communities can lose their permanence if a predator or a competitor is removed: the remaining species no longer coexist. This well-known phenomenon is analysed for some low dimensional examples of Lotka-Volterra type, with special attention paid to the occurrence of heteroclinic cycles.     

Non-equilibrium thermodynamics of Lotka-Volterra ecosystems: Stability and evolution

A non-equilibrium thermodynamic theory of generalized Lotka-Volterra ecosystem has been presented. The main results consist of the derivation of a generalized expression of entropy-production for the evolutionary ecosystem and the study of its role in the analysis of ecological stability, succession and also in the formulation of some extremum principles characterising the evolution of the ecosystem.     

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.     

Species assembly in model ecosystems, I: Analysis of the population model and the invasion dynamics

Recently we have introduced a simplified model of ecosystem assembly (Capitán et al., 2009) for which we are able to map out all assembly pathways generated by external invasions in an exact manner. In this paper we provide a deeper analysis of the model, obtaining analytical results and introducing some approximations which allow us to reconstruct the results of our previous work. In particular, we show that the population dynamics equations of a very general class of trophic-level structured food-web have an unique interior equilibrium point which is globally stable. We show analytically that communities found as end states of the assembly process are pyramidal and we find that the equilibrium abundance of any species at any trophic level is approximately inversely proportional to the number of species in that level. We also find that the per capita growth rate of a top predator invading a resident community is key to understand the appearance of complex end states reported in our previous work. The sign of these rates allows us to separate regions in the space of parameters where the end state is either a single community or a complex set containing more than one community. We have also built up analytical approximations to the time evolution of species abundances that allow us to determine, with high accuracy, the sequence of extinctions that an invasion may cause. Finally we apply this analysis to obtain the communities in the end states. To test the accuracy of the transition probability matrix generated by this analytical procedure for the end states, we have compared averages over those sets with those obtained from the graph derived by numerical integration of the Lotka-Volterra equations. The agreement is excellent.     

n种群Lotka-Volterra时滞系统的全局稳定性，Ⅲ:必要性

本文考虑无对角时滞的ｎ种群Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ时滞系统,证明了弱对角占优（ＷＤＤ）为系统关于所有时滞全局稳定的充要条件。     

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．