Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.     

n维可投影LOtka-Volterra竞争系统的渐近性

对于二维和三维的Lotka-Volterra竞争系统,已有文献证明:当每一个坐标轴上的平衡点均为渐近稳定时,该系统几乎所有解趋于坐标轴上平衡点所组成的点集,即,不趋于坐标轴上平衡点的解集,其测度为零.由此, van den Driessche和Zeeman于1998年提出猜测:对n(n>3)维Lotka-Volterra竞争系统,当每一个坐标轴上的平衡点均为渐近稳定时,该系统几乎所有解趋于坐标轴上平衡点所组成的点集,即,不趋于坐标轴上平衡点的解集,其在n维空间的测度为零.本文证明当n维Lotka-Volterra竞争系统可被逐维投影到一维系统时,该猜测成立,并给出了可投影条件的代数判据.本文所得结论包含了已有文献的结果.     

Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems

We consider the geometry of carrying simplices of discrete-time competitive Kolmogorov systems. An existence theorem for the carrying simplex based upon the Hadamard graph transform is developed, and conditions for when the transform yields a sequence of convex or concave graphs are determined. As an application it is shown that the planar Leslie–Gower model has a carrying simplex that is convex or concave.     

A particle swarm pattern search method for bound constrained global optimization

In this paper we develop, analyze, and test a new algorithm for the global minimization of a function subject to simple bounds without the use of derivatives. The underlying algorithm is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the optional search phase of pattern search we apply a particle swarm scheme to globally explore the possible nonconvexity of the objective function. Our extensive numerical experiments showed that the resulting algorithm is highly competitive with other global optimization methods also based on function values. Support for Luís N. Vicente was provided by Centro de Matemática da Universidade de Coimbra and by FCT under grant POCI/MAT/59442/2004.     

Topological properties of strongly monotone planar vector fields

We consider strongly monotone continuous planar vector fields with a finite number of fixed points. The fixed points fall into three classes, attractors, repellers and saddles. Naturally, the relative positions of the fixed points must obey a set of restrictions imposed by monotonicity. The study of these restrictions is the main goal of the paper. With any given vector field, we associate a matrix describing the arrangement of the fixed points on the plane. We then use these matrices to formulate simple necessary and sufficient conditions which allow one to determine whether a finite set of attractors, repellers and saddles at given positions on the plane can be realized as the fixed point set of a strongly monotone vector field.     

Convergence analysis of a monotone projection algorithm in reflexive banach spaces

In this article, fixed points of generalized asymptotically quasi-φ-nonexpansive mappings and equilibrium problems are investigated based on a monotone projection algorithm. Strong convergence theorems are established without the aid of compactness in the framework of reflexive Banach spaces.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

On evolutionary ray-projection dynamics

We introduce the ray-projection dynamics in evolutionary game theory by employing a ray projection of the relative fitness (vector) function, i.e., a projection unto the unit simplex along a ray through the origin. Ray-projection dynamics are weakly compatible in the terminology of Friedman (Econometrica 59:637–666, 1991), each of their interior fixed points is an equilibrium and each interior equilibrium is one of its fixed points. Furthermore, every interior evolutionarily stable strategy is an asymptotically stable fixed point, and every strict equilibrium is an evolutionarily stable state and an evolutionarily stable equilibrium. We also employ the ray-projection on a set of functions related to the relative fitness function and show that several well-known evolutionary dynamics can be obtained in this manner.     

Strong Convergence Theorems for Variational Inequalities and Fixed Point Problems of Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a modified hybrid Mann iterative scheme with perturbed mapping which is based on well-known CQ method, Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this modified hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.     

连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性

本文研究任意有限维空间中连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性.借助适当的线性变换和坐标变换,将局部稳定流形和不稳定流形拉直,利用奇异流映射和正则流映射构造了Poincaré映射.通过技巧性地估计向量的模,给出了在横截面上Poincaré映射的初始点与首次回归点离异宿轨道与横截面交点的距离之比,得到了高维空间中连接两个带有一维不稳定流形的异宿环的非常简洁的稳定性判据.     

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known.     

A wide neighborhood primal-dual predictor-corrector interior-point method for symmetric cone optimization

Our aim in this paper is to introduce a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverse-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. Some strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As an application, we give an algorithm to solve fixed point problems for nonexpansive mappings, variational inequality problems and equilibrium problems in Hilbert spaces. Finally, we give one numerical example to illustrate our convergence analysis.     

On the finite-dimensional dynamical systems with limited competition

The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.

     

On multidimensional discrete-time Beverton–Holt competition models

Following Ackleh et al. (2005), we study the multidimensional discrete-time competitive Beverton–Holt equations with equal interspecific competition coefficients. It is shown that competitive exclusion occurs if only one species has the largest carrying capacity. Otherwise, all the species with the largest carrying capacity coexist. In the former case, the system is globally asymptotically stable. In the latter case, the system has a linear stable manifold.     

食饵带疾病的捕食模型的全局稳定性

本文研究了一类三维生态传染病模型的正解性和边界性,并分析了系统平衡点的局部稳定性。利用一种新的几何方法,获得了内平衡点的全稳定性,推广了Li和Muldowney[1]提出的这种方法的应用,这种方法避免了寻找Lyapunov的困难。     

Convergence Theorems for Pointwise Asymptotically Strict Pseudo-Contractions in Hilbert Spaces

A new class of contractive mappings called pointwise asymptotically ?-strict pseudo-contractions in Hilbert spaces is introduced and weak convergence of the sequence generated by Mann's iterative scheme to a fixed point of a uniformly Lipschitzian and pointwise asymptotically ?-strict pseudo-contractive mapping T in a Hilbert space is established. Also, a new kind of monotone hybrid method which is a modification of Mann's iterative scheme for finding a common fixed point of an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ?-strict pseudo-contractive mappings is proposed. Strong convergence of the sequence generated by the proposedmonotone hybrid method for an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ?-strict pseudo-contractive mappings in a Hilbert space is also shown. The results presented in this article extend and improve some known results in the literature.     

Strong convergence of an iterative method for pseudo-contractive and monotone mappings

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.     

Carrying simplices in nonautonomous and random competitive Kolmogorov systems

The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system.     

On a sharpened form of the Leray-Lions' ellipticity criterion

The well-known ellipticity criterion of Leray-Lions for solvability of a nonlinear equation in divergence form is sharpened, eliminating any coercivity hypothesis, by using a remark on the asymptotic strong monotonicity of strictly monotone mappings in finite dimensional spaces, and extending the result to non-local mappings.