一类退化与非退化稳态扩散模型正解的存在性

考虑一类退化与非退化的稳态扩散模型。利用上、正解方法与Schauder不动点原理，证明此类稳态扩散模型在一定条件下正解的存在性。     

一类具有扩散和交错扩散的捕食模型的正解

讨论了带扩散和交错扩散的三种群捕食模型.应用上下解方法,得到这类捕食模型正解的存在性,同时研究了其正解的不存在性.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

捕食者和食饵均带有扩散的随机捕食-食饵模型动力学分析

考虑了斑块环境下捕食者种群和食饵种群分别在n个斑块扩散的随机捕食 食饵模型.利用Lyapunov函数法证明了对任意给定的初始值,随机系统全局正解的存在唯一性,并对其进行了有界性分析.此外给出了食饵种群及整个系统灭绝的充分条件.最后通过数值模拟验证了所得理论的正确性.     

一类具有交叉扩散的捕食模型非常数正解的存在性

研究了一类具有扩散和交叉扩散项的Holling-Tanner捕食-食饵模型.首先利用最大值原理和Harnack不等式给出正解的先验估计,进一步利用度理论得到非常数正解的存在性与不存在性,从而给出非常数正解存在的充分条件.     

带有自由边界的反应扩散系统解的整体存在与爆破（英文）

本文研究一类带耦合非线性反应项的反应扩散系统的自由边界问题.为简便起见,假设条件和解都是径向对称的.首先,利用压缩映射定理,给出正解的局部存在性和唯一性.然后,考虑解的爆破性质和长时间行为.     

具有连续时滞和基于比率的非自治捕食扩散系统的持续生存

研究具有连续时滞和基于比率的非自治捕食扩散系统.证明了该系统一致持久性及任何正解全局渐近稳定性的充分条件.     

一类带有交叉扩散的捕食-食饵模型的正解

该文研究了一类在齐次Dirichlet边界条件下的带有交叉扩散的捕食-食饵模型.首先,根据Leray-Schauder度理论,建立了系统的正解的存在性;其次,当参数m=且充分大时,分别研究了正则扰动方程和奇异扰动方程的正解的存在性,和借助分歧理论说明奇异系统的正解在a~*处爆破;最后,建立了系统正解的多解性.     

具有非线性收获的Leslie-Gower捕食者-食饵扩散模型的Hopf分支分析

研究一类具有非线性收获和扩散的Leslie-Gower捕食者-食饵模型.通过对常微分系统和反应扩散系统产生Hopf分支条件的讨论,分析收获和扩散在系统产生Hopf分支中的作用.     

光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.     

Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects

A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. By using the implicit function theorem and the Lyapunov-Schmidt reduction method, the existence of the positive solutions bifurcating from the trivial solution is obtained. Furthermore, the stability of the bifurcating positive solutions is also investigated by analyzing the associated characteristic equation.     

Stability Analysis of an Enterprise Competitive Model with Time Delay

A three-dimensional enterprise competitive model with time delay is considered. Where the delay is regarded as bifurcation parameters. By analyzing the corresponding characteristic equation of positive equilibrium,the local stability of positive equilibrium is regarded. By using the normal form method and center manifold theorem, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are shown to illustrate the obtained results.     

Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.     

A stage-structured predator-prey system with time delay

A stage-structured predator-prey system with time delay is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.     

Global topological perturbations of nonlinear elliptic eigenvalue problems

This paper is concerned with bifurcation from infinity for nonlinear elliptic equations, which are not necessarily linearizable at infinity. The methods employed are global perturbation techniques by means of which one obtains access to continua of positive solutions bifurcating from infinity via continua bifurcating from trivial solutions.     

Hopf bifurcation analysis in a tri-neuron network with time delay

A neural network model with three neurons and a single delay is considered. The existence of local Hopf bifurcations is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. A global Hopf bifurcation theorem due to Wu and a Bendixson's criterion for high-dimensional ODE due to Li and Muldowney are used to obtain a group of conditions for the system to have multiple periodic solutions when the delay is sufficiently large. Finally, numerical simulations are carried out to support the theoretical analysis of the research.     

Hopf bifurcation analysis for a delayed predator-prey system with a prey refuge and selective harvesting

In this paper, a delayed predator-prey system with Holling type III functional response incorporating a prey refuge and selective harvesting is considered. By analyzing the corresponding characteristic equations, the conditions for the local stability and existence of Hopf bifurcation for the system are obtained, respectively. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given.     

Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.     

LOCAL AND GLOBAL HOPF BIFURCATIONS FOR A PREDATOR-PREY SYSTEM WITH TWO DELAYS

In this paper, the Leslie predator-prey system with two delays is studied. The stability of the positive equilibrium is discussed by analyzing the associated characteristic transcendental equation. The direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory. The conditions to guarantee the global existence of periodic solutions are given.     

含两时滞捕食-被捕食系统的稳定性及分歧

本文研究一类含两相异时滞的捕食－被捕食系统的稳定性及分歧。首先，我们讨论两相异时滞对系统唯一正平衡点的稳定性的影响，通过对系数与时滞有关的特征方程的分析，建立了一种稳定性判别性。其次，将一个时滞看成分歧参数，而另一个看作固定参数，我们证明了该系统具有HOPF分歧特性。最后，我们讨论了分歧解的稳定性。