Propagation of Local Disturbances in Reaction Diffusion Systems Modeling Quadratic Autocatalysis

This article studies the propagation of initial disturbance in a quadratic autocatalytic chemical reaction in one-dimensional slab geometry, where two chemical species A, called the reactant, and B, called the autocatalyst, are involved in the simple scheme A + B - \u3e 2B. Experiments demonstrate that chemical systems for which quadratic or cubic catalysis forms a key step can support propagating chemical wavefronts. When the autocatalyst is introduced locally into an expanse of the reactant, which is initially at uniform concentration, the developing reaction is often observed to generate two wavefronts, which propagate outward from the initial reaction zone. We show rigorously that with such an initial setting the spatial region is divided into three regions by the two wavefronts. In the middle expanding region, the reactant is almost consumed so that A approximate to 0, whereas in the other two regions there is basically no reaction so that B approximate to 0. Most of the chemical reaction takes place near the wavefronts. The detailed characterization of the concentrations is given for each of the three zones

Steady states and dynamics of an autocatalytic chemical reaction model with decay

The dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered. Nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates, upper-lower solution method, and bifurcation theory. The effects of decay order, decay rate and diffusion rates to the dynamical behavior are discussed. (c) 2012 Published by Elsevier Inc

Computational Study of Traveling Wave Solutions and Global Stability of Predator-Prey Models

In this thesis, we study two types of reaction-diffusion systems which have direct applications in understanding wide range of phenomena in chemical reaction, biological pattern formation and theoretical ecology. The first part of this thesis is on propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order $m$ without decay. The second is chemical reaction of order $m$ with a decay of order $l$, where $m$ and $l$ are positive integers and $m \u3e l\ge1$. A typical system is $A + 2B \rightarrow3B$ and $B\rightarrow C$ involving three chemical species, a reactant A and an auto-catalyst B and C an inert chemical species. We use numerical computation to give more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves. For autocatalytic reaction of order $m = 2$ with linear decay $l = 1$, which has a particular important role in biological pattern formation, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters. The second part of this thesis is on the global stability of diffusive predator-prey system of Leslie Type and Holling-Tanner Type in a bounded domain $\Omega\subset R^N$ with no-flux boundary condition. By using a new approach, we establish much improved global asymptotic stability of a unique positive equilibrium solution. We also show the result can be extended to more general type of systems with heterogeneous environment and/or other kind of kinetic terms