Je Chiang Tsai

Existence and stability of traveling waves in buffered systems

We study wave propagation in the buffered bistable equation, i.e., the bistable equation where the diffusing species reacts with immobile buffers that restrict its diffusion. Such a model describes wave front propagation in excitable systems where the diffusing species is buffered; in particular, the study of the propagation of waves of increased calcium concentration in a variety of cell types depends directly upon the analysis of such buffered excitability. However, despite the biological importance of these types of equations, there have been almost no analytical studies of their properties. Here, we study the question of whether or not the inclusion of multiple buffers can eliminate propagated waves. First, we prove that a unique (up to translation) traveling wave front exists. Moreover, the wave speed is also unique. Then we prove that this traveling wave front is stable, i.e., that any initial condition which vaguely resembles a traveling wave front (in a way we make precise) evolves to the unique wave front. We thus prove that multiple stationary buffers cannot prevent the existence of a traveling wave front in the buffered bistable equation and may not eliminate stable wave fronts. This suggests (although we do not prove) that the same result is true for more complex and realistic models of calcium wave propagation, a result of direct physiological relevance.

Traveling Waves in the Buffered FitzHugh–Nagumo Model

In many physiologically important excitable systems, such as intracellular calcium dynamics, the diffusing variable is highly buffered. In addition, all physiological buffered excitable systems contain multiple buffers, with different affinities. It is thus important to understand the properties of wave solutions in excitable systems with multiple buffers, and to understand how multiple buffers interact. Under the assumption that buffering acts on a fast time scale, we derive a criterion for the existence of a traveling pulse with positive wave speed in the buffered FitzHugh–Nagumo model, a prototypical excitable system. This condition suggests that there exists a critical excitability corresponding to the excitability parameter

Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

a_c

The Structure of Solutions for a Third Order Differential Equation in Boundary Layer Theory

such that, for systems with excitability above this critical excitability (the excitability parameter

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

a\in(0,a_c)

Similarity solutions for boundary layer flows with prescribed surface temperature

), buffers cannot prevent the propagation of traveling pulses with positive wave speed, provided that the parameter

On the steadily rotating spirals

\epsilon\ll1

Wave propagation in predator-prey systems

. Further, buffers can speed u...

Traveling waves in a simplified model of calcium dynamics

We study the existence and uniqueness of traveling wave solutions for a class of twocomponent reaction diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor model, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinburgh 2005; 135A:663–675), but also establishes the existence and uniqueness of traveling wave solutions to the reaction-diffusion system for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.

Asymptotic stability of traveling wave fronts in the buffered bistable system

In this paper, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.