Class Of Reaction-diffusion Systems Research Articles

Stability of a Planar Front in a Class of Reaction-Diffusion Systems

We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front t...

Dissipative reaction diffusion systems with quadratic growth

We introduce a class of reaction diffusion systems of which weak solution exists global-in-time with relatively compact orbit in L 1. Reaction term in this class is quasi-positive, dissipative, and up to with quadratic growth rate. If the space dimension is less than or equal to two, the solution is classical and uniformly bounded. Provided with the entropy structure, on the other hand, this weak solution is asymp-totically spatially homogeneous.

Computational Study of Traveling Wave Solutions of Isothermal Chemical Systems

AbstractThis article studies 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 n, where m and n are positive integers and m>n≥1. A typical system in autocatalysis is A+2B→3B and B→C involving two chemical species, a reactant A and an auto-catalyst B and C an inert chemical species.The numerical computation gives 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 n = 1, which has a particular important role in chemical waves, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters.

Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems

The transverse stability of localized stripe patterns for certain singularly perturbed two‐component reaction‐diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi‐strong interaction regime, the cross‐sectional profile of the stripe is well‐approximated by a homoclinic pulse solution of the corresponding 1‐D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself‐adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Gierer–Meinhardt model and for a three‐component RD system modeling patterns of criminal activity in urban crime.

On existence of wavefront solutions in mixed monotone reaction-diffusion systems

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.

Auxiliary field loop expansion of the effective action for a class of stochastic partial differential equations

We present an alternative to the perturbative (in coupling constant) diagrammatic approach for studying stochastic dynamics of a class of reaction diffusion systems. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions of the fields describing these systems. The systems we consider include Langevin systems describable by the set of self interacting classical fields ϕi(x,t) in the presence of external noise ηi(x,t), namely (∂t−ν∇2)ϕ−F[ϕ]=η, as well as chemical reaction annihilation processes obtained by applying the many-body approach of Doi–Peliti to the Master Equation formulation of these problems. We consider two different effective actions, one based on the Onsager–Machlup (OM) approach, and the other due to Janssen–deGenneris based on the Martin–Siggia–Rose (MSR) response function approach. For the simple models we consider, we determine an analytic expression for the Energy landscape (effective potential) in both formalisms and show how to obtain the more physical effective potential of the Onsager–Machlup approach from the MSR effective potential in leading order in the auxiliary field loop expansion. For the KPZ equation we find that our approximation, which is non-perturbative and obeys broken symmetry Ward identities, does not lead to the appearance of a fluctuation induced symmetry breakdown. This contradicts the results of earlier studies.

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

Uniform Boundedness and Decay Estimates for a System of Reaction-Diffusion Equations with a Triangular Diffusion Matrix on $${\mathbb{R}^{n}}$$

This paper concerns with a class of reaction-diffusion systems with triangular diffusion matrix on the unbounded domain \({\mathbb{R}^{n}}\) . The system with diagonal diffusion matrix has been studied by J. D. Avrin and F. Rothe in [4]. We prove two new results about uniform boundedness to solutions of such class of reaction-diffusion systems in \({BUC(\mathbb{R}^{n})}\), the space of bounded uniformly continuous functions from \({\mathbb{R}^{n}}\) to \({\mathbb{R}}\) .

A passivity-based stability criterion for reaction diffusion systems with interconnected structure

In this paper, stability of a class of reaction diffusion systems is studied. Conditions on global asymptotic stability of the homogeneous equilibrium are derived based on the diagonal stability of a dissipativity matrix. This work extends previous result on global asymptotic stability from cyclic systems to general systems with interconnected structure. In addition, it reformulates the approach using an input-output formalism that makes the results easier to understand and apply. A biological example from the Mitogen-Activated Protein Kinase (MAPK) system is provided at the end to illustrate the new approach and the main result.

Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models

This paper deals with the global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts in a class of reaction-diffusion systems. The known results do not apply in solving these problems because the reaction terms do not satisfy the required monotone condition. To overcome the difficulty, a weak monotone condition is proposed for the reaction terms, which is called interval monotone condition. Under such a weak monotone condition, the existence and comparison theorem of solutions is first established for reaction-diffusion systems on R by appealing to the theory of abstract differential equations. The global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts are then proved by the elementary super- and sub-solution comparison and squeezing methods for nonlinear evolution equations. Finally, these abstract results are applied to a two species competition-diffusion model and a system modeling man–environment–man epidemics.

Pattern selection and control via localized feedback

Many theoretical analyses of feedback control of pattern-forming systems assume that feedback is applied at every spatial location, something that is often difficult to accomplish in experiments. This paper considers an experimentally more feasible scenario where feedback is applied at a sparse array of discrete spatial locations. We show how such feedback can be computed analytically for a class of reaction-diffusion systems and use generalized linear stability analysis to determine how dense the actuator array needs to be to select or maintain control of a given pattern state in the presence of noise. The one-dimensional Swift-Hohenberg equation is used to illustrate our theoretical results and explain earlier experimental observations on the control of the Rayleigh-Bénard convection.

Higher Dimensional SLEP Equation and Applications to Morphological Stability in Polymer Problems

Existence and stability of stationary internal layered solutions to a rescaled diblock copolymer equation are studied in higher dimensional space. Rescaling is necessary since the characteristic domain size of any stable pattern eventually vanishes in an appropriate singular limit. A general sufficient condition for the existence of singularly perturbed solutions and the associated stability criterion are given in the form of linear operators acting only on the limiting location of the interface. Applying the results to radially symmetric and planar patterns, we can show, for instance, stability of radially symmetric patterns when one of the components of diblock copolymer dominates the other, and that of the long-striped pattern in a long and narrow domain for the planar case. These results are consistent with the experimental ones. The above existence and stability criterion can be easily extended to a class of reaction diffusion systems of activator-inhibitor type.

Chemical turbulence equivalent to Nikolavskii turbulence.

We find evidence that a certain class of reaction-diffusion (RD) systems can exhibit chemical turbulence equivalent to Nikolaevskii turbulence. We study an extended complex Ginzburg-Landau (CGL) equation derived from this class of RD systems. First, we show numerically that the power spectrum of this CGL equation, in the neighborhood of a codimension-two Turing-Benjamin-Feir point, is qualitatively quite similar to that of the Nikolaevskii equation. Then, we demonstrate that the Nikolaevskii equation can in fact be obtained from this CGL equation through a phase reduction procedure.

Asymptotic behavior of a competition-diffusion system with time delays

In this paper, we study a class of reaction-diffusion systems modelling the dynamics of two competing species with time delay effects. The global asymptotic convergence is established in terms of the rate constants of reaction function, independent of the time delays and the effect of diffusion by the upper-lower solutions and iteration method.

Convergence Rates for a Reaction-Diffusion System

A class of reaction-diffusion systems is investigated. This class is motivated by some diffusive epidemic models, which serve to modelise the spread of Feline Immunodeficiency Virus (FIV) in the cat population, and sexually transmitted diseases. We obtain exponential convergence rates for a system with unbounded time dependent coefficients.

Phases of a conserved mass model of aggregation with fragmentation at fixed sites

To study the effect of quenched disorder in a class of reaction-diffusion systems, we introduce a conserved mass model of diffusion and aggregation in which fragmentation occurs only at certain fixed sites. On most sites, the mass moves as a whole to a nearest neighbor while it leaves the fixed sites only as a single monomer (i.e., chips off). Once the mass leaves any site, it coalesces with the mass present on its neighbor. We study in detail the effect of a single chipping site on the steady state in arbitrary dimensions, with and without bias. In the thermodynamic limit, the system can exist in one of the following three phases. (a) Pinned aggregate (PA) phase in which an infinite aggregate (with mass proportional to the volume of the system) appears at the chipping site with probability one but not in the bulk. (b) Unpinned aggregate (UA) phase in which the infinite aggregate occurs at the chipping site with a probability strictly less than one and can coexist with infinite aggregates in the bulk. (c) Nonaggregate (NA) phase in which there is no infinite cluster. The steady state of the system depends on the dimension and drive. A sitewise inhomogeneous mean field theory predicts that the system exists in the UA phase in all cases. Monte Carlo simulations in one and two dimensions support this prediction in all but one-dimensional, biased case. In the latter case, there is a phase transition from the NA phase to the PA phase as the density is increased. We identify the critical point exactly and calculate the mass distribution in the PA phase. The NA phase and the critical point are studied by Monte Carlo simulations and using scaling arguments. A variant of the above aggregation model is also considered in which total particle number is conserved and chipping occurs at a fixed site, but the particles do not interact with each other at other sites. This model is solved exactly by mapping it to a zero range process. With increasing density, it exhibits a phase transition from the NA phase to the PA phase in all dimensions, irrespective of bias. The free-particle model is also solved with an extensive number of chipping sites with random chipping rates and we argue that it qualitatively describes the behavior of the aggregation model with extensive disorder.

Modeling chemical reactions in rivers: A three component reaction

What follows is the analysis of a model for dynamics of chemical reactions in a river. Dominant forces to be considered include diffusion, advection, and rates of creation or destruction of participating species (due to chemical reactions). In light of this, the model we will be using will be based upon a nonlinear system of reaction-advection-diffusion equations. The nonlinearity comes solely from the influences of the chemical reactions. First, we will establish some general results for reaction diffusion systems. In particular, we will illustrate a class of reaction diffusion systems whose solutions are bounded from below by zero. We will also provide a local existence result for this class of problems. Afterwards, we will focus on the dynamics of an equidiffusive three component reaction system. Specifically, we will provide conditions under which one could be guaranteed the existence of global solutions. We will also discuss the qualities of the $\omega$-limit set for this system.

Lyapunov functions and Lp-estimates for a class of reaction-diffusion systems

We give a sufficient condition for the existence of a Lyapunov function for the system $$\eqalign{ a_t&=\nabla(k(a,c)\nabla a-h(a,c)\nabla c),\quad\ x\in{\mit\Omega} ,\ t>0,\cr \varepsilon c_t&=k_c{\mit\Delta} c-f(c)c+g(a,c),\quad\ x\in{\mit\Omega} ,\ t>0

Persistence of zero velocity fronts in reaction diffusion systems.

Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets of control parameters. When varying one control parameter, the front velocity may become zero only at isolated values (where the Maxwell condition is satisfied, for potential systems). The experimental observation of fronts with a zero velocity over a finite interval of parameters, e.g., in catalytic experiments [Barelko et al., Chem. Eng. Sci., 33, 805 (1978)], therefore, seems paradoxical. We show that the velocity dependence on the control parameter may be such that velocity is very small over a finite interval, and much larger outside. This happens in a class of reaction diffusion systems with two components, with the extra assumptions that (i) the two diffusion coefficients are very different, and that (ii) the slowly diffusing variables has two stable states over a control parameter range. The ratio of the two velocity scales vanishes when the smallest diffusion coefficient goes to zero. A complete study of the effect is carried out in a model of catalytic reaction. (c) 2000 American Institute of Physics.

On the persistence of spatiotemporal oscillations generated by invasion

Many systems in biology and chemistry are oscillatory, with a stable, spatially homogeneous steady state which consists of periodic temporal oscillations in the interacting species, and such systems have been extensively studied on infinite or semi-infinite spatial domains. We consider the effect of a finite domain, with zero-flux boundary conditions, on the behaviour of solutions to oscillatory reaction-diffusion equations after invasion. We begin by considering numerical simulations of various oscillatory predator-prey systems. We conclude that when regular spatiotemporal oscillations are left in the wake of invasion, these die out, beginning with a decrease in the spatial frequency of the oscillations at one boundary, which then propagates across the domain. The long-time solution in this case is purely temporal oscillations, corresponding to the limit cycle of the kinetics. Contrastingly, when irregular spatiotemporal oscillations are left in the wake of invasion, they persist, even in very long time simulations. To study this phenomenon in more detail, we consider the λ-ω class of reaction-diffusion systems. Numerical simulations show that these systems also exhibit die-out of regular spatiotemporal oscillations and persistence of irregular spatiotemporal oscillations. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the transition fronts in r and θ x which occur during the die-out of the regular oscillations. We then use this approximation to describe how the die-out occurs, and to derive a measure of its rate, as a function of parameter values. We discuss aplications of our results to ecology, calcium signalling and chemistry.