一类具有时滞及反馈控制的非自治非线性比率依赖食物链模型

该文研究了一类具有时滞和反馈控制的三种群非线性非自治比率依赖的食物链模型.首先,基于时滞微分不等式理论,提出了一些新的分析方法,并构造了一个合适的李亚普诺夫函数.其次,得到了系统正解的持久性和全局吸引性的充分条件.第三,利用理论分析和不动点理论,讨论了相应的周期系统,建立了周期系统正周期解的存在性、唯一性和稳定性的充分条件.另外,给出了一些数值模拟,证明了我们的理论分析是正确的.最后,给出了相应的具有乘法噪声源的随机食物链模型的数值例子,并得到该模型一些新的有趣的解的变化过程.

ON A MULTI-DELAY LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH FEEDBACK CONTROLS AND PREY DIFFUSION

This article is focusing on a class of multi-delay predator-prey model with feedback controls and prey diffusion. By developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the globally attractivity of positive solution for the predator-prey system.Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system.

On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals

For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

On a delay ratio-dependent predator-prey system with feedback controls and shelter for the prey

In this paper,a class of three-species multi-delay Lotka-Volterra ratio-dependent predator-prey model with feedback controls and shelter for the prey is considered.A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator-prey system are established by developing some new analysis methods and using the theory of differentim inequalities as well as constructing a suitable Lyapunov function.Furthermore,some conditions for the existence,uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques.In addition,some numerical solutions of the equations describing the system are given to show that the obtained criteria are new,general,and easily verifiable.Finally,we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources,and obtain some new interesting dynamical behaviors of the system.At the same time,the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator-prey models.

Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work(in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.