This talk will cover some recent progress on numerical homotopy method to solve systems of nonlinear partial differential equations (PDEs) arising from biology and physics. This new approach, which is used to compute multiple solutions and bifurcation of nonlinear PDEs, makes use of polynomial systems (with thousands of variables) arising by discretization. Examples from hyperbolic systems, tumor growth models, and a blood clotting model will be used to demonstrate the ideas.

 
Chemotaxis is the biased motion of cells under the influnce of chemicals that attract the cells.
The most important phenomenon about chemotaxis is cell aggregation, for which we use non- constant, especially spiky or transition-layer (step function-like) steady states to model. In the case of 1D spatial domains, we present two methods to establish the existence of such steady states: (i) global bifurcation theory combined with Helly's compactness theorem and Sturm oscillation theorem; (ii) singular perturbation method. We also prove local asymptotic stability and uniqueness of these steady states

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on joint work with Gui-Qiang Chen.

 
Reaction-diffusion model is one of the attractive models used to study pattern formation in different biological systems, from individual cell components to developing tissues. In this talk, I will introduce several reaction-diffusion models arising from the studies of cell polarization and tissue pattern formation.
In the first part, I will present a two-equation reaction-diffusion model for studying cell polarization, which is central to carry out processes such as differentiation, migration and development. I will perform linear stability analysis, in particular Turing stability analysis to the model to derive conditions of parameters for which cell polarity emerges without any spatial cue. I will apply live cell imaging and mathematical modeling to understand how diploid daughter cells in budding yeasts establish polarity preferentially at the pole distal to the previous division site.
In the second part, I will introduce several morphogen-mediated patterning models to study how tissue pattern formation is robust to the environmental noises and perturbations. Morphogens are important signaling molecules governing the pattern formation of multicellular organisms during embryo development. In this talk, I will discuss how two mechanisms, the expansion-repression mechanism and presence of non-signaling receptors, play a role for overcoming the effect of the fluctuations in morphogen and receptor production rates.