The institute is dedicated in the mathematical problems arising in different disciplines including physics, ecology and atmospheric sciences et al. In particular, we are interested in dynamics of differential equations in these fields and the illustrations of natural phenomena by these conclusions. Our main research branches include: infinite dimensional dynamical systems, differential equations and dynamical systems, mathematical ecology, Partial differential equations in fluid dynamics.

●Differential Equations and Dynamical Systems:This group is under the supervision of Professor Wan-Tong Li. The following is their recent advances. 1) Dr Zhi-Cheng Wang, Wan-Tong Li and Shigui Ruan established the existence, uniqueness and asymptotic stability of traveling wave solutions of reaction-diffusion systems with nonlocal delays, and formulated the effect of nonlocal delays; Dr. Guo Lin, Wan-Tong Li and their collaborators studied the traveling wave solutions of competitive systems formulating the evolutionary of several competitors. 2) Dr Zhi-Cheng Wang, Wan-Tong Li and theircollaboratorsestablished new entire solutions of reaction-diffusion systems with nonlocal delays and reflected some new dynamical properties of these systems. These solutions are defined for all temporal variable, which are useful in understanding the structure of global attractors and reflecting the instantaneous dynamics of these systems. 3) Dr. Guo Lin, Wan-Tong Li and theircollaboratorsestimated the asymptotic speed of spreading of some non-cooperative systems. These conclusions describe the role of coupled nonlinearities of some systems. 4). Dr. Bin-Guo Wang, Wan-Tong Li developed the theory of skew product semiflows applicable to some non-monotone systems. 5). Dr. Li and his collaborators considered the dynamics of nonlocal dispersal systems. They established the existence, uniqueness and asymptotic stability of traveling wave solutions by using the theory of monotone dynamical systems. They then constructed different/new entire solutions of nonlocal dispersal models by studying the precise properties of traveling wave solutions. Recently, they also investigated the asymptotic spreading of nonlocal dispersal equations with degenerate nonlinearities. 6). Dr. Zhi-Cheng Wangstudied the existence, uniqueness and stability of traveling curved fronts for reaction-diffusion bistable systems in two dimensional space. His results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of n obligate mutualist. Dr. Wang andhis collaborators also considered two-dimensional curved fronts and high-dimensional pyramidal traveling fronts in time-periodic reaction-diffusion equations with bistable nonlinearity, and estalished the existence, uniqueness and stability of these nonplanar traveling fronts.

●Infinite Dimensional Dynamical Systems: Using some ideas from the theory of nonlinear functional analysis, we are dedicated to the study of the long-time behaviour of the infinite dimensional dynamical systems (generated by parabolic partial differential equations, hyperbolic partial differential equations, solitary equations, lattice differential equations, delay differential equations, and stochastic differential equations). This is a challenging problem, since it can provide useful information on the future evolution of the system. Recently, we are mainly concerned with the theory of the existence, regularity and structure of attractors, and the application in a number of evolutionary equations arising from problems in physical and life sciences. Some interesting results have been published by “Indiana Univ. Math. J.”, “J. Diff. Equations”，“Trans. Amer. Math. Soc.”, “Discrete Contin. Dyn. Syst.”, “Quart. Appl. Math.” and “Set-Valued Anal.”, etc. And the project “The long-time behavior and steady state solution of the infinite dimensional dynamical systems” gained the first level prize of “Gansu provincial progress prize in scientific and collective technology” in 2007.

●Mathematical Ecology:Mathematical Ecology research covers a broad variety of topics ranging from theoretical issues of ecosystem's complexity, chaos, noise, pattern formation, metapopulation, evaluation of ecosystem risk, ecological effects and dynamical consequence to practical agricultural applications. Our main focused field:1.The group studied the microbial biomass and activity in alkalized magnesic soils under arid conditions. The paper was published by Soil Biology & Biochemistry, which is a top journal in ecology. 2.The group applied reliability theory in an arid area and evaluated oasis ecosystem risk of the Shiyang River Basin, China. The results could be used as a guild in controlling and managing ecosystem risk in the research area.3. The relationship on the stochastic ecosystem resilience and productivity were found and the results suggested that external stochastic perturbation, partly, if not all, responsible for the high variations of primary productivity of ecosystems, was a significant one affecting resilience-productivity relationship. 4. The dynamical complexity and metapopulation persistence were investigated and the results indicated that the overcrowding effect is the key to incur chaos and increase the dynamical complexity and to improve the persistence of metapopulations. 5. The three-level trophic food chain with consumer mutual interference (MIF) was studied and the results served to provide insight into the effects of MIF in the real world. 6. The group researched the effect of the environmental noise, dispersal and spatial heterogeneity on the spectral color of a spatially-structured population. One of the importance of our study is enriching the theory of reddened spectrum in natural world. 7. The group studied the effects of prey refuges on a class of predator-prey system and found that the destabilizing effect did not occur in Gause type system. Base on the research in mathematical ecology, our group awarded the following foundation: 1.The National Social Science Fund Foundation of China (No.99BJY048, 04AJL007). 2.The National Natural Science Fund Foundation of China (No.39970135, 2002CCA00300, 30470298, 30870478, 11126183).

●Partial Differential Equations in Fluid Dynamics: Fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.The main areas of our research include non-linear dynamical systems, non- Newtonian fluid equations, stochastic fluid equations. We are concerned about the qualitative study of solutions of incompressible viscous fluid equations and stochastic fluid equations.(By May,7,2012)