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Research Projects

My research interests span the areas of numerical computation and applied analysis in mathematical biology and physics. In particular, my work is motivated by understanding the deterministic models in different scales and their connections.

Mathematical Biology

Tumor Growth
Joint project with physicists to develop a continuous model for tumor growth. In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or/and on the mechanical stress in the tumor. The individual based model (IBM) has already been tested in accordance with experiments. Quantitative matching can be found by comparing the numerical results of the continuous model and IBM.
For the two by two degenerate parabolic/elliptic reaction-diffusion system model, we prove that there exist traveling waves above a minimal speed, and analyze their shapes. In particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation.
Modeling and Analysis for Chemotactic Movements.
  • We created a new class of hyperbolic Keller-Segel model (HKS) which can predict branching instabilities of the bacteria Bacillus subtilis on nutrient rich media, then proved the existence of its steady state and traveling wave solutions. The key contribution of HKS is its different branching mechanisms from all previous models and without using local nutrient depletion.

  • Simulations of self-propelled chemotactic bacteria in a stokes flow. The particles moving in the fluid represent bacteria of the E. Coli type which interact with their chemical environment through consumption of nutriments and orientation in some favorable direction. By following the trace of each bacteria and looking at their group behavior, we are able to reproduce some phenomena in the biology experiments.
  • Derivation of a new moment system of a pathway-based mean-field theory (PBMFT) using the moment closure technique in kinetic theory. By incorporating the most recent quantitatively measured signaling pathway, we are able to explain a counter-intuitional experimental observation which showed that in a spatial-temporal fast-varying environment, there exists a phase shift between the dynamics of ligand concentration and center of mass of the cells.
  • Numerical Methods

    AP Method
    The multiscale problems and problems with interfaces are important since they arise in many physical applications. One way to solve those problems is to develop Asymptotic Preserving (AP) schemes. For PDEs with small parameters, a scheme is AP if it possesses the discrete analogy of the continuous asymptotic limit as the small parameter goes to zero. AP methods are attractive since they work for all range of corresponding parameter and the uniform convergence.
  • From Neutron Transport Equation to its Diffusion Limit.
  • From Compressible Euler Equation to its Incompressible Low Mach Number Limit.
  • From the Complex Ginzburg-Landau and nonlinear Schr\"odinger equation to its Large Time and Space Scale Limit.
  • Relaxation method for traveling wave simulations
    There is a vast number of biological phenomena where the key elements or precursors to a development process exhibit traveling waves. Classical models describing these phenomena include various reaction-diffusion equations. More recent models motivated by biological phenomena lead to non-local PDEs or systems with singularities. The relaxation method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. It is easy to implement, and it applies to classical differential equations as well as nonlocal equations and systems with singularities.
  • The traveling wave solution of the nonlocal Fisher equation that connects two unstable steady states.
  • One dimensional traveling wave solutions for non-local PDEs or systems with singularitie.
  • One dimensional pulsating traveling front for periodic advection diffusion reaction equations.