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We consider a priori estimates of the Weyl's embedding problem of $(/mathbb{S}^2, g)$ in general $3$-dimensional Riemannian manifold $(N^3, /bar g)$. We establish mean curvature estimate under natural geometric assumption. Together with a recent work by Li-Wang, we obtain an isometric embedding of $(/mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl's isometric embedding theorem in space form under the condition that $g/in C^2$ with $D^2g$ Dini continuous.

In this talk, we study a reaction-diffusion population model with time delay. We establish a comparison principle for coupled upper/lower solutions and prove the existence/uniqueness result for the model. We then show the global asymptotic behavior of the model.

In this talk, I will present some results on chemostat models. In particular, I will report some progress on competitive exclusion principle. In addition, I will also present some results on factors leading to transient oscillations in chemostat.

Mathematical biologists apply mathematical, computational and experimental techniques to investigate biological processes at all levels, from the molecular to the population. To setup an approach spanning multiple levels of biophysical reality, I will describe a new 3D hybrid framework which includes three major submodels: the discrete stochastic Subcellular Element Model for individual cellular activity; continuum reaction-diffusion-advection partial differential equations of extracellular biomolecules; a stochastic decision-making model for cell lineage transition.

We study the existence and nonexistence of traveling waves of diffusive epidemic models. The model systems are non-monotone because the disease models exhibit a predator-prey mechanism. We will construct a suitable convex set in a weighted function space, and then apply Schauder fixed point theorem. It turns out that the basic reproduction number of the corresponding ordinary differential equations plays an important role in the existence theory of traveling waves.

I first talk about the importance of phage therapy in fighting bacteria infections and review the advances of mathematical modeling in the study of phage dynamics. Then, I introduce our researches on how immune responses affect the outcomes of phage therapy. It is shown that the host immune response induces the backward bifurcation.

In this talk, we present a method to analyze certain fast-slow motions in dynamical systems. For singular perturbed dynamical systems, the well-known Geometric Singular Perturbation Technique (GSPT) is usually applied to find the special limit cycles -- fast-slow periodic solutions. However, many practical problems may not be able to be put in the form of singular perturbed equations, but they still exhibit fast-slow motions. In that case, based on dynamical system theory

In this talk, I will give the Agmon-type estimates of Bergman kernels under the conditions of the curvature or the Ricci curvature bounds. This is joint work with Shoo Seto