Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

Integrability of continuous bundles

We give new sufficient conditions for the integrability and unique integrability of continuous tangent subbundles on manifolds of arbitrary dimension, generalizing Frobenius’ classical theorem for {C^{1}} subbundles. Using these conditions, we derive new criteria for uniqueness of solutions to ODEs and PDEs and for the integrability of invariant bundles in dynamical systems. In particular, we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).