Abstract:
We study front propagation in the irreversible epidemic model $A+B\to 2A$ in one dimension. Here, we allow the particles $A$ and $B$ to diffuse with rates $D_A$ and $D_B$, which, in general, may be different. We find analytic estimates for the front velocity by writing truncated master equation in a frame moving with the rightmost $A$ particle. The results obtained are in reasonable agreement with the simulation results and are amenable to systematic improvement. We also observe a crossover from the linear dependence of front velocity $V$ on $D_A$ for smaller values of $D_A$ to $V\propto \sqrt{D_A}$ for larger $D_A$, but numerically still significantly different from the mean field value. The deviations reflect the role of internal fluctuations which is neglected in the mean field description.

Abstract:
We study front propagation in the reaction diffusion process $\{A\stackrel{\epsilon}\to2A, A\stackrel {\epsilon_t}\to3A\}$ on a one dimensional (1d) lattice with hard core interaction between the particles. Using the leading particle picture, velocity of the front in the system is computed using different approximate methods, which is in good agreement with the simulation results. It is observed that in certain ranges of parameters, the front velocity varies as a power law of $\epsilon_t$, which is well captured by our approximate schemes. We also observe that the front dynamics exhibits temporal velocity correlations and these must be taken care of in order to find the exact estimates for the front diffusion coefficient. This correlation changes sign depending upon the sign of $\epsilon_t-D$, where $D$ is the bare diffusion coefficient of $A$ particles. For $\epsilon_t=D$, the leading particle and thus the front moves like an uncorrelated random walker, which is explained through an exact analysis.

Abstract:
We consider an individual-based two-dimensional spatial model with nearest-neighbor preemptive competition to study front propagation between an invader and a resident species. In particular, we investigate the asymptotic front velocity and compare it with mean-field predictions.

Abstract:
we study a discrete model of the irreversible autocatalytic reaction a + b ？ 2a in one dimension. looking at the dynamics of propagation, we find that in the low-concentration limit the average velocity of propagation approaches v = q/2, where q is the concentration, and, in the high concentration limit, we nd the velocity approaches v = 1 - e-q/2.

Abstract:
We study a discrete model of the irreversible autocatalytic reaction A + B -> 2A in one dimension. Looking at the dynamics of propagation, we find that in the low-concentration limit the average velocity of propagation approaches v = theta/2, where theta is the concentration, and, in the high concentration limit, we nd the velocity approaches v = 1 - e-theta/2.

Abstract:
Recent theoretical work has shown that so-called pulled fronts propagating into an unstable state always converge very slowly to their asymptotic speed and shape. In the the light of these predictions, we reanalyze earlier experiments by Fineberg and Steinberg on front propagation in a Rayleigh-B\'enard cell. In contrast to the original interpretation, we argue that in the experiments the observed front velocities were some 15% below the asymptotic front speed and that this is in rough agreement with the predicted slow relaxation of the front speed for the time scales probed in the experiments. We also discuss the possible origin of the unusually large variation of the wavelength of the pattern generated by the front as a function of the dimensionless control parameter.

Abstract:
This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v*, the asymptotic rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v*, and pushed fronts, nonlinear fronts whose asymptotic speed v^dagger is larger than v*. In addition, this approach allows us to clarify many aspects of the front selection problem, the question whether for a given dynamical equation the front is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v*. Almost half of the paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable states from this unified perspective. The paper also includes short sections on the derivation of the universal power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts, on the relation between so-called global modes and front propagation, and on stochastic fronts.

Abstract:
The propagation of a front connecting a stable homogeneous state with a stable periodic state in the presence of additive noise is studied. The mean velocity was computed both numerically and analitically. The numerics are in good agreement with the analitical prediction.

Abstract:
The problem of front propagation in flowing media is addressed for laminar velocity fields in two dimensions. Three representative cases are discussed: stationary cellular flow, stationary shear flow, and percolating flow. Production terms of Fisher-Kolmogorov-Petrovskii-Piskunov type and of Arrhenius type are considered under the assumption of no feedback of the concentration on the velocity. Numerical simulations of advection-reaction-diffusion equations have been performed by an algorithm based on discrete-time maps. The results show a generic enhancement of the speed of front propagation by the underlying flow. For small molecular diffusivity, the front speed $V_f$ depends on the typical flow velocity $U$ as a power law with an exponent depending on the topological properties of the flow, and on the ratio of reactive and advective time-scales. For open-streamline flows we find always $V_f \sim U$, whereas for cellular flows we observe $V_f \sim U^{1/4}$ for fast advection, and $V_f \sim U^{3/4}$ for slow advection.

Abstract:
We consider the propagation of a flame front in a solid medium with a periodic structure. The model is governed by a free boundary system for the pair " temperature-front. " The front's normal velocity depends on the temperature via a (degenerate) Arrhenius kinetic. It also depends on the front's mean curvature. We show the existence of travelling wave solutions for the full system and consider their homogenization as the period tends to zero. We analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the limiting ratio " curvature coefficient/period. " This analysis provides valuable information on the heterogeneous propagation as well.