Local bifurcation from the second eigenvalue of the Laplacian in a square

In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point . We also compute the Morse index of the solutions in the four branches.

     

Bifurcation in the stable manifold of a chemostat with general polynomial yields

A three‐dimensional chemostat with nth‐ and mth‐order polynomial yields, instead of the particular ones such as A+BS, A+BS², A+BS³, A+BS⁴, A+BS² + CS³, and A+BSⁿ, is proposed. The existence of limit cycles in the two‐dimensional stable manifold, the Hopf bifurcation, and the stability of the periodic solution created by the bifurcation is proved. Copyright © 2009 John Wiley & Sons, Ltd.     

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.     

具有质载和内抑制剂的非均匀恒化器模型共存解的多重性

本文研究一类具有质载和内抑制剂的非均匀恒化器竞争模型. 基于对第二类极限系统分歧的进一步分析, 我们研究了此模型共存解的多重性和稳定性, 从而在一定条件下得到了模型共存解分歧的确切形状, 验证了已有的数值结果.     

双营养物的非均匀Chemostat模型解的性质

本文讨论双营养物的非均匀Chemostat模型解的性质.通过运用极值原理,上下解方法以及分歧理论得到了正平衡解的存在性,运用稳定性理论证明了正平衡解的稳定性.     

Bifurcation for a functional yield chemostat when one competitor produces a toxin

A model, with general yield functions: Fi(S), i=1,2, of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper. The conditions in terms of the relevant parameters for the Hopf bifurcation of the three-dimensional system have been proved, which implies the existence of limit cycles in the 3-D system.     

一类捕食——食饵模型正平衡解的整体分歧

讨论了一类改进的Leslie-Gower和Holling-Type Ⅱ型捕食-食饵模型对应的平衡态系统正解的结构.以捕食者的出生率b为分歧参数,利用局部分歧理论和整体分歧理论,得到了此平衡态系统正解的存在性与参数b的关系,即当b适当大时,该平衡态系统具有共存正解.     

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

Dynamics and steady-state analysis of an unstirred chemostat model with internal storage and toxin mortality

This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.     

A continuation method for a weakly elliptic two-parameter eigenvalue problem

We show that the continuation method can be used to solve aweakly elliptic two-parameter eigenvalue problem. We generalizethe continuation method for a nonsymmetric eigenvalue problemAx = x by T. Y. Li, Z. Zeng and L. Cong (1992 SIAM J. Numer.Anal. 29, 229–248) to two-parameter problems.     

Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

This paper is devoted to the study of global bifurcation from infinity of nontrivial solutions of a nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. We prove the existence of two families of unbounded continua of nontrivial solutions to this problem, which emanate from bifurcation points in $ℝ\times \{\infty \}$ and possess oscillatory properties of eigenfunctions (and their derivatives) of the corresponding linear problem in some neighborhoods of these bifurcation points.     

具有饱和项的互惠系统的分歧与稳定性

运用谱分析和分歧理论的方法,在齐次Dirichlet边界条件下,对具有饱和项的互惠系统的非负定态解的分歧及其稳定性进行研究.一方面,分别以生长率作为分歧参数,讨论了发自半平凡解的分歧;另一方面,以两物种的生长率作为分歧参数,利用Liapunov-Schmidt过程,研究了在二重特征值处的分歧;同时判定了这些分歧解的稳定性.     

含分散时滞反馈chen系统的分支分

研究了含分散时滞反馈的Chen系统,利用Routh-Hurwitz准则分析了在弱核及强核情形下平衡点的局部稳定性及Hopf分支的存在性．还运用规范型理论及中心流形定理,得出了包括决定分支周期解的方向、稳定性和周期的清晰的计算公式,其结果可用于混沌控制分析．     

Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.     

非线性算子的渐近歧点

利用拓扑度方法作为主要工具研究Banach空间中非线性算子的渐近歧点的存在性,但并没有假定非线性算子是渐近线性算子.最后,把一般结果应用于常微分方程的Sturm-Liouville问题.     

一类捕食者-食饵模型的稳定性及Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支.以滞量为参数,得到了系统正平衡点的稳定性和Hopf分支存在的充分条件.利用规范型和中心流形定理,给出了确定Hopf分支方向和分支周期解的稳定性的计算公式.     

Stability and Hopf Bifurcation of Stationary Solution of a Delay Equation

In this paper we investigate a Logistic equation with delay and it is shown that if b(x) > c(x), the stationary solution is globally asymptotically stable; if τ is small, U(x) is locally stable; if b(x) < c(x). there is Hopf bifurcation from U(x).     

多时滞合作系统的稳定性与全局Hopf分支

以时滞为参数,研究了一类多时滞合作系统的正平衡点的稳定性及局部Hopf分支的存在性.在此基础上结合一般泛函微分方程的全局Hopf分支定理,讨论了该系统全局Hopf分支的存在性.