# Influence of self- and cross-diffusion on wave train solutions of reaction-diffusion systems

10 Jun 2005-International Journal of Systems Science (Taylor & Francis Group)-Vol. 36, Iss: 7, pp 415-422

TL;DR: The present paper deals with the wave train solutions of a generalized reaction-diffusion system and a two-species oscillatory model has been considered as an example and analysed for its spiral wave solutions.

Abstract: The present paper deals with the wave train solutions of a generalized reaction-diffusion system. A comparative study of the nature of waves of the system in the presence of both self-diffusion and cross-diffusion has been performed. A two-species oscillatory model has also been considered as an example and analysed for its spiral wave solutions.

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TL;DR: The critical wave speed required for the existence of wave solutions connecting the trivial with the nontrivial equilibrium has been found out and shown to depend on different system parameters together with the dispersal rate.

Abstract: A three variable mathematical model describing the propagation of an infectious disease in a human population is proposed and analyzed. The human population is assumed to live in two distinct habitats with no inter-habitat migration. The infectious agent disperse randomly among the said habitats. Methods of upper and lower solutions are used to establish the existence of traveling wave solutions connecting the trivial with the nontrivial equilibrium. The critical wave speed required for the existence of such wave solutions has been found out and shown to depend on different system parameters together with the dispersal rate.

43 citations

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Williams College

^{1}TL;DR: This work extends previous work on ideal replicators to include the square root rate and other possible nonlinearities, which it couple with an enzymatic sink, and obtains exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations.

Abstract: Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.

36 citations

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Williams College

^{1}TL;DR: In this article, the effect of cross-diffusion in pattern formation has been investigated for the Templator model of chemical self-replication, and exact relations between the relevant parameters for the Poincare-Adronov-Hopf and Turing bifurcations have been obtained.

Abstract: For the Templator model of chemical self-replication, we consider the effect of cross-diffusion in pattern formation. Although other chemical systems with cross-diffusion have been reported, we are the first to obtain exact relations between the relevant parameters for the Poincare–Adronov–Hopf and Turing bifurcations. Moreover, for the simplest case, we obtain analytical expressions in parameter space for the bifurcation curves and the Turing–PAH codimension two point.

29 citations

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01 Oct 1989

TL;DR: The Poincare-Bendixson Theorem as mentioned in this paper describes the existence, uniqueness, differentiability, and flow properties of vector fields, and is used to prove that a dynamical system is Chaotic.

Abstract: Equilibrium Solutions, Stability, and Linearized Stability * Liapunov Functions * Invariant Manifolds: Linear and Nonlinear Systems * Periodic Orbits * Vector Fields Possessing an Integral * Index Theory * Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows * Asymptotic Behavior * The Poincare-Bendixson Theorem * Poincare Maps * Conjugacies of Maps, and Varying the Cross-Section * Structural Stability, Genericity, and Transversality * Lagrange's Equations * Hamiltonian Vector Fields * Gradient Vector Fields * Reversible Dynamical Systems * Asymptotically Autonomous Vector Fields * Center Manifolds * Normal Forms * Bifurcation of Fixed Points of Vector Fields * Bifurcations of Fixed Points of Maps * On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution * The Smale Horseshoe * Symbolic Dynamics * The Conley-Moser Conditions or 'How to Prove That a Dynamical System is Chaotic' * Dynamics Near Homoclinic Points of Two-Dimensional Maps * Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields * Melnikov's Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields * Liapunov Exponents * Chaos and Strange Attractors * Hyperbolic Invariant Sets: A Chaotic Saddle * Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems * Global Bifurcations Arising from Local Codimension-Two Bifurcations * Glossary of Frequently Used Terms

4,945 citations

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368 citations

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TL;DR: In this paper, the authors considered the solution of reaction-diffusion systems near a Hopf bifurcation point and showed that the one-armed spiral wave is probably stable and the multi-armed wave is unstable for small values of the parameter q.

Abstract: We consider the reaction-diffusion system \[\begin{gathered} R_T =
abla ^2 R + R\left( {1 - R^2 - \vec
abla \theta \cdot \vec
abla \theta } \right), \hfill \\ R\theta _T = R
abla ^2 \theta + 2\vec
abla R \cdot \vec
abla \theta + qR^3 \hfill \\ \end{gathered} \]This system governs the solutions of reaction-diffusion systems near a Hopf bifurcation point. In two spatial dimensions we use formal asymptotic expansions to contruct one-armed and multi-armed Archimedean spiral waves for small values of the parameter q. We then show that the one-armed spiral waves are probably stable and the multi-armed ones are unstable for $| q |$ small. Next, by numerical continuation methods, we construct spiral waves for all q. These calculations show that the one-armed spiral waves are unstable when $| q | > 1.397 \cdots $. We also find the explicit one-dimensional analogues of spiral waves for all q, and show that they are unstable for $| q |$ small.

300 citations

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266 citations

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