Abstract:
We study front propagation and diffusion in the reaction-diffusion system A $\leftrightharpoons$ A + A on a lattice. On each lattice site at most one A particle is allowed at any time. In this paper, we analyze the problem in the full range of parameter space, keeping the discrete nature of the lattice and the particles intact. Our analysis of the stochastic dynamics of the foremost occupied lattice site yields simple expressions for the front speed and the front diffusion coefficient which are in excellent agreement with simulation results.

Abstract:
We consider an irreversible autocatalytic conversion reaction A+B -> 2A under subdiffusion described by continuous time random walks. The reactants' transformations take place independently on their motion and are described by constant rates. The analog of this reaction in the case of normal diffusion is described by the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation leading to the existence of a nonzero minimal front propagation velocity which is really attained by the front in its stable motion. We show that for subdiffusion this minimal propagation velocity is zero, which suggests propagation failure.

Abstract:
Propagation of traveling fronts in three-dimensional reaction-diffusion media with spatially modulated cross-section is studied using the Schl\"ogl model as a representative example. Applying appropriate perturbation techniques leads first to a reduction of dimensionality in which the spatially dependent Neumann boundary condition translate into a boundary-induced advection term and, secondly, to an equation of motion for the traveling wave position in weakly corrugated confinements. Comparisons with numerical simulations demonstrate that our analytical results properly predicts the nonlinear dependence of the propagation velocity on ratio of the spatial period of the confinement to the intrinsic width of the front; including the peculiarity of propagation failure. Based on the eikonal equation, we obtain an analytical estimate for the finite interval of propagation failure. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants if the intrinsic width of the front is much larger than the spatial variation of the medium.

Abstract:
We investigate the influence of fluid flows on the propagation of chemical fronts arising in FKPP type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large P\'eclet ($Pe$) and Damk\"ohler ($Da$) numbers. The front speed is expressed in terms of a periodic path -- an instanton -- that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for $(\log Pe)^{-1}\ll Da\ll Pe$ and for $Da\gg Pe$. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection--diffusion--reaction equation.

Abstract:
A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that combines fractional diffusion and regular diffusion. The reaction kinetics corresponds to a Fisher-Kolmogorov nonlinearity. The numerical method is based on a finite-difference operator splitting algorithm with an explicit Euler step for the time advance of the reaction kinetics, and a Crank-Nicholson semi-implicit time step for the transport operator. The anomalous diffusion operators are discretized using an upwind, flux-conserving, Grunwald-Letnikov finite-difference scheme applied to the regularized fractional derivatives. With fractional diffusion of order $\alpha$, fronts exhibit exponential acceleration, $a_L(t) \sim e^{\gamma t/\alpha}$, and develop algebraic decaying tails, $\phi \sim 1/x^{\alpha}$. In the case of tempered fractional diffusion, this phenomenology prevails in the intermediate asymptotic regime $\left(\chi t \right)^{1/\alpha} \ll x \ll 1/\lambda$, where $1/\lambda$ is the scale of the tempering. Outside this regime, i.e. for $x > 1/\lambda$, the tail exhibits the tempered decay $\phi \sim e^{-\lambda x}/x^{\alpha+1}$, and the front velocity approaches the terminal speed $v_*= \left(\gamma-\lambda^\alpha \chi\right)/ \lambda$. Of particular interest is the study of the interplay of regular and fractional diffusion. It is shown that the main role of regular diffusion is to delay the onset of front acceleration. In particular, the crossover time, $t_c$, to transition to the accelerated fractional regime exhibits a logarithmic scaling of the form $t_c \sim \log \left(\chi_d/\chi_f\right)$ where $\chi_d$ and $\chi_f$ are the regular and fractional diffusivities.

Abstract:
We consider a reaction-diffusion equation in narrow random channels. We approximate the generalized solution to this equation by the corresponding one on a random graph. By making use of large deviation analysis we study the asymptotic wave front propagation.

Abstract:
We expand on a previous study of fronts in finite particle number reaction-diffusion systems in the presence of a reaction rate gradient in the direction of the front motion. We study the system via reaction-diffusion equations, using the expedient of a cutoff in the reaction rate below some critical density to capture the essential role of fl uctuations in the system. For large density, the velocity is large, which allows for an approximate analytic treatment. We derive an analytic approximation for the front velocity depe ndence on bulk particle density, showing that the velocity indeed diverge s in the infinite density limit. The form in which diffusion is impleme nted, namely nearest-neighbor hopping on a lattice, is seen to have an essential impact on the nature of the divergence.

Abstract:
We study the propagation of pulled fronts in the $A <-> \leftrightarrow A+A$ microscopic reaction-diffusion process using Monte Carlo (MC) simulations. In the mean field approximation the process is described by the deterministic Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation. In particular we concentrate on the corrections to the deterministic behavior due to the number of particles per site $\Omega$. By means of a new hybrid simulation scheme, we manage to reach large macroscopic values of $\Omega$ which allows us to show the importance in the dynamics of microscopic pulled fronts of the interplay of microscopic fluctuations and their macroscopic relaxation.

Abstract:
Propagating fronts arising from bistable reaction-diffusion equations are a purely deterministic effect. Stochastic reaction-diffusion processes also show front propagation which coincides with the deterministic effect in the limit of small fluctuations (usually, large populations). However, for larger fluctuations propagation can be affected. We give an example, based on the classic spruce-budworm model, where the direction of wave propagation, i.e., the relative stability of two phases, can be reversed by fluctuations.

Abstract:
We study the front propagation in Reaction-Diffusion systems whose reaction dynamics exhibits an unstable fixed point and chaotic or noisy behaviour. We have examined the influence of chaos and noise on the front propagation speed and on the wandering of the front around its average position. Assuming that the reaction term acts periodically in an impulsive way, the dynamical evolution of the system can be written as the convolution between a spatial propagator and a discrete-time map acting locally. This approach allows us to perform accurate numerical analysis. They reveal that in the pulled regime the front speed is basically determined by the shape of the map around the unstable fixed point, while its chaotic or noisy features play a marginal role. In contrast, in the pushed regime the presence of chaos or noise is more relevant. In particular the front speed decreases when the degree of chaoticity is increased, but it is not straightforward to derive a direct connection between the chaotic properties (e.g. the Lyapunov exponent) and the behaviour of the front. As for the fluctuations of the front position, we observe for the noisy maps that the associated mean square displacement grows in time as $t^{1/2}$ in the pushed case and as $t^{1/4}$ in the pulled one, in agreement with recent findings obtained for continuous models with multiplicative noise. Moreover we show that the same quantity saturates when a chaotic deterministic dynamics is considered for both pushed and pulled regimes.