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

Stable foliations near a traveling front for reaction diffusion systems

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front

Anomalous Wave Dispersion in the Cyclohexanedione-Bromate Chemical Oscillator

A modified six-variable Oregonator model presented here successfully reproduces a significant portion of the behavior observed in the Ferroin-catalyzed cyclohexanedione variant of the Belousov-Zhabotinsky (CHD-BZ) reaction. The phenomena of anomalous velocity dispersion (in which following waves may catch up to, rather than fall behind an initial excitation wave), wave-stacking, and backfiring have been successfully reproduced numerically as resulting from non-monotonic [Br-] decay to the steady state in the wake of an excitation pulse. The non-monotonic decay is seen as a dip in [Br-] following the passage of a chemical wave. This dip in [Br-] decay curve allows a following wave to accelerate and catch up to the initial wave. The origin of anomalous dispersion as the result of such a non-monotonic decay curve in [Br-] has been suggested previously by Steinbock et al. and Szalai et al. However, the work presented here is the first successful representation of anomalous wave-velocity dispersion using a chemical model. This model is based on the well-understood chemistry of the Oregonator model of the Belousov-Zhabotinsky reaction, coupled to a second pathway (based on chemistry related to uncatalyzed bromate oscillators) for the oxidation of organic substrate to provide the new dynamics

Computing stability of multi-dimensional travelling waves

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions.Comment: 23 pages, 9 figures (some in colour). v2: added details and other changes to presentation after referees' comments, now 26 page

Stability of traveling waves in partly parabolic systems

Abstract. We review recent results on stability of traveling waves in partly parabolic reactiondiffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave

Numerical methods for the travelling wave solutions in reaction-diffusion equations

In this work we consider how shooting and relaxation methods can be used to investigate propagating waves solutions of PDEs. Particular attention is paid to overcoming some of the numerical difficulties. The linear stability of these solutions are analyzed by using the Evans function approach. As an illustration, we shall apply the above methods to an autocatalytic reaction involving two diffusing chemicals in one spatial dimension. Prospects of further work are also discussed

Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects

Kinetic Intersections as a Source of Information on Rate Coefficients

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