New results on the global asymptotic stability of?a?Lotka-Volterra system

In this paper, we revisit the famous Lotka-Volterra competitive system. By combining spectral matrix theory with Lyapunov function, some new sufficient conditions are obtained to guarantee the global asymptotic stability of a unique equilibrium for Lotka-Volterra competitive system. Our new results generalize and significantly improve the known results in the previous literature. The main purpose of this paper is to propose a new methodology to study the high-dimensional Lotka-Volterra system. And this method can be extensively used to study the global asymptotic stability of the equilibrium. Finally, some examples and their simulations show the feasibility and effectiveness of our results.     

模糊随机时滞Lotka-Volterra模型的持久性和渐近性

给出模糊随机时滞Lotka-Volterra模型,通过Ito公式,在一定条件下研究模型(1.2)的随机持久性,利用指数鞅不等式进一步给出了解的渐近估计.最后,通过两个数值算例对主要结果进行验证.     

Least squares data fitting with implicit functions

This paper discusses the computational problem of fitting data by an implicitly defined function depending on several parameters. The emphasis is on the technique of algebraic fitting off(x, y; p) = 0 which can be treated as a linear problem when the parameters appear linearly. Various constraints completing the problem are examined for their effectiveness and in particular for two applications: fitting ellipses and functions defined by the Lotka-Volterra model equations. Finally, we discuss geometric fitting as an alternative, and give examples comparing results.     

周期系数三种群Lotka－Volterra混合模型分析

考虑三种群Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ周期系数模型，种群间既有捕食关系又有竞争关系，得到唯一存在全局渐近稳定周期解的条件，并举例说明条件的可行性．     

Coexistence and Asymptotic Periodicity in a Competition Model of Plankton Allelopathy

In this paper, the two species Lotka-Volterra competition model of plankton allelopathy from aquatic ecology is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions and self-diffusions are weak. The existence of the positive T-periodic solutions and the global stability as well as the global attractivity for the parabolic system are also given.     

A diffusive predator-prey model in heterogeneous environment

In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Lotka-Volterra equations in three dimensions satisfying the Kowalevski-Painlevé property

We examine a class of Lotka-Volterra equations in three dimensions which satisfy the Kowalevski-Painlevé property. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painlevé analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient.     

三维Lotka-Volterra 系统正拟周期解的存在性和稳定性

This work focuses on the existence and stability of positive quasi-periodic solutions for the 3-dimensional Lotka-Volterra system. Using KAM （Kolmogorov-Arnold-Moser） theory and Newton iteration, it is shown that there exists a positive quasi-periodic solution in a Cantor family for the 3-dimensional Lotka-Volterra system. On the above basis, we can show the stability of the solution with the help of Lyapunov function.     

一类多偏差变元的n种群Lotka-Volterra模型的周期正解

本文研究了一类多偏差变元Lotka-Volterra种群模型的间期正解问题,利用重合度拓展定理和一些分析技巧,得到了周期正解存在性的新结果.与已有文献相比,本文所讨论的模型更具一般性,它包含了以前人们所研究的竞争-种群模型、捕食-种群模型等,而且估计先验界的方法也是全新的.     

一类随机模型的最优捕获问题研究

在Volterra两种群竞争模型的基础上,构造了随机的具有捕获的两种群竞争模型,研究讨论了捕获对种群生长过程的影响和如何实现最优捕获等问题.从确定性模型入手,深入讨论随机竞争模型的收获最优问题.通过对捕获强度E和贴现率等的估计与讨论,计算出了最优捕获强度最优捕获量最优经济收益.     

From behavioural to population level: Growth and competition

The aim of this work is to build models of population dynamics for growth and competition interaction by starting with detailed models at the individual level. At the individual level, we start with detailed models where the growth is described by linear terms. By considering individual interferences and by using aggregation methods, we show that the population level, different growth equation can emerge. We present an example of the emergence of logistic growth and an example of the emergence of logistic growth with Allee effect. Furthermore, in the case of two populations, we show that individual interferences can lead at the population level, to a model which has the same qualitative dynamics behaviour as the Lotka-Volterra competition model. Finally, we show that our model brings to light the effects of spatial heterogeneity on competition models. First, we find the stabilizing effects but also we show that destabilizing effects can occur.     

Stochastic Lotka-Volterra system with infinite delay

This paper investigates a stochastic Lotka-Volterra system with infinite delay, whose initial data comes from an admissible Banach space C_r. We show that, under a simple hypothesis on the environmental noise, the stochastic Lotka-Volterra system with infinite delay has a unique global positive solution, and this positive solution will be asymptotic bounded. The asymptotic pathwise of the solution is also estimated by the exponential martingale inequality. Finally, two examples with their numerical simulations are provided to illustrate our result.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

The Center Conditions and Hopf Cyclicity for a 3D Lotka-Volterra System

The main objective of this paper is not only to find necessary and The main objective of this paper is not only to find necessary and sufficient conditions for the existence of a center on a local center manifold for a three dimensional Lotka-Volterra system, but also to determine the maximum number of limit cycles that can bifurcate from the positive equilibrium as a fine focus. Firstly, the singular point quantities are computed and simplified to obtain necessary conditions for local integrability, and Darboux method is applied to show the sufficiency. Then, the Hopf bifurcation on the center manifold is investigated, from this, the conclusion of at most five small limit cycles generated in the vicinity of the equilibrium is obtained. To the best of our knowledge, this is the first case with five possible limit cycles for the cyclicity of 3D Lotka-Volterra systems.     

The method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction-diffusion systems

The aim of this paper is to show the existence of solutions of the n-dimensional diffraction problem for weakly coupled quasilinear elliptic reaction-diffusion system. The coefficients of the equations under consideration are allowed to be discontinuous. We extend the method of upper and lower solutions for reaction-diffusion equations with continuous coefficients to the elliptic diffraction problem. An application of these results is given to the steady-state problem of Lotka-Volterra cooperation model with two cooperating species.     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling wave solutions for a Lotka-Volterra reaction-diffusion competition model with time delay. By constructing upper and lower solutions, we obtain the precise minimum wave speed of traveling waves under certain conditions. Our results also extend the known results on the minimum wave speed for Lotka-Volterra competition model without delay.     

A time-fractional competition ecological model with cross-diffusion

This paper is concerned with some mathematical and numerical aspects of a Lotka-Volterra competition time-fractional reaction-diffusion system with cross-diffusion effects. First, we study the existence of weak solutions of the model following the well-known Faedo-Galerkin approximation method and convergence arguments. We demonstrate the convergence of approximate solutions to actual solutions using the energy estimates. Next, the Galerkin finite element scheme is proposed for the considered model. Further, various numerical simulations are performed to show that the fractional-order derivative plays a significant role on the morphological changes of the considered competition model.     

中立型时滞Lotka-Volterra系统正周期解的存在性

本文研究了具有m个捕食者n个食饵的中立型时滞Lotka-Volterra系统,并 利用重合度理论中的连续性引理,得到了该系统正周期解存在性的充分条件.