Fluctuations and stability of fisher waves

Shape of population interfaces as an indicator of mutational instability in coexisting cell populations

Cellular populations such as avascular tumors and microbial biofilms may 'invade' or grow into surrounding populations. The invading population is often comprised of a heterogeneous mixture of cells with varying growth rates. The population may also exhibit mutational instabilities, such as a heavy deleterious mutation load in a cancerous growth. We study the dynamics of a heterogeneous, mutating population competing with a surrounding homogeneous population, as one might find in a cancerous invasion of healthy tissue. We find that the shape of the population interface serves as an indicator for the evolutionary dynamics within the heterogeneous population. In particular, invasion front undulations become enhanced when the invading population is near a mutational meltdown transition or when the surrounding 'bystander' population is barely able to reinvade the mutating population. We characterize these interface undulations and the effective fitness of the heterogeneous population in one- and two-dimensional systems.

Macroscopic response to microscopic intrinsic noise in three-dimensional Fisher fronts

We study the dynamics of three-dimensional Fisher fronts in the presence of density fluctuations. To this end we simulate the Fisher equation subject to stochastic internal noise, and study how the front moves and roughens as a function of the number of particles in the system, N. Our results suggest that the macroscopic behavior of the system is driven by the microscopic dynamics at its leading edge where number fluctuations are dominated by rare events. Contrary to naive expectations, the strength of front fluctuations decays extremely slowly as 1/logN, inducing large-scale fluctuations which we find belong to the one-dimensional Kardar-Parisi-Zhang universality class of kinetically rough interfaces. Hence, we find that there is no weak-noise regime for Fisher fronts, even for realistic numbers of particles in macroscopic systems.

Speed of invasion of an expanding population by a horizontally transmitted trait

Range expansions are a ubiquitous phenomenon, leading to the spatial spread of genetic, ecological, and cultural traits. While some of these traits are advantageous (and hence selected), other, nonselected traits can also spread by hitchhiking on the wave of population expansion. This requires us to understand how the spread of a hitchhiking trait is coupled to the wave of advance of its host population. Here, we use a system of coupled Fisher-Kolmogorov-Petrovsky-Piskunov (F-KPP) equations to describe the spread of a horizontally transmitted hitchhiking trait within a population as it expands. We extend F-KPP wave theory to the system of coupled equations to predict how the hitchhiking trait spreads as a wave within the expanding population. We show that the speed of this trait wave is controlled by an intricate coupling between the tip of the population and trait waves. Our analysis yields a new speed selection mechanism for coupled waves of advance and reveals the existence of previously unexpected speed transitions.

Spontaneous propagation of self-assembly in a continuous medium

We report a mechanism in which self-assembly propagates spontaneously in a continuous medium, enabling the delivery of local order information to distance. In a large stable system a locally self-assembled structure as a precursor destabilizes its surrounding areas through a dipole interaction. The newly formed structures inherit the same order information from the precursor and further activate the self-assembly of their neighbors. This process causes spatial extension of self-assembly and replication of the order, producing extremely long-range ordered superlattice without defects.

Spatial variability enhances species fitness in stochastic predator-prey interactions

We study the influence of spatially varying reaction rates on a spatial stochastic two-species Lotka-Volterra lattice model for predator-prey interactions using two-dimensional Monte Carlo simulations. The effects of this quenched randomness on population densities, transient oscillations, spatial correlations, and invasion fronts are investigated. We find that spatial variability in the predation rate results in more localized activity patches, which in turn causes a remarkable increase in the asymptotic population densities of both predators and prey and accelerated front propagation.

Front propagation in A+B-->2A reaction under subdiffusion

We consider an irreversible autocatalytic conversion reaction A+B-->2A under subdiffusion described by continuous-time random walks. The reactants' transformations take place independently of 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 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.

Propagating waves of self-assembly in organosilane monolayers

Wavefronts associated with reaction-diffusion and self-assembly processes are ubiquitous in the natural world. For example, propagating fronts arise in crystallization and diverse other thermodynamic ordering processes, in polymerization fronts involved in cell movement and division, as well as in the competitive social interactions and population dynamics of animals at much larger scales. Although it is often claimed that self-sustaining or autocatalytic front propagation is well described by mean-field "reaction-diffusion" or "phase field" ordering models, it has recently become appreciated from simulations and theoretical arguments that fluctuation effects in lower spatial dimensions can lead to appreciable deviations from the classical mean-field theory (MFT) of this type of front propagation. The present work explores these fluctuation effects in a real physical system. In particular, we consider a high-resolution near-edge x-ray absorption fine structure spectroscopy (NEXAFS) study of the spontaneous frontal self-assembly of organosilane (OS) molecules into self-assembled monolayer (SAM) surface-energy gradients on oxidized silicon wafers. We find that these layers organize from the wafer edge as propagating wavefronts having well defined velocities. In accordance with two-dimensional simulations of this type of front propagation that take fluctuation effects into account, we find that the interfacial widths w(t) of these SAM self-assembly fronts exhibit a power-law broadening in time, w(t) approximately t(beta), rather than the constant width predicted by MFT. Moreover, the observed exponent values accord rather well with previous simulation and theoretical estimates. These observations have significant implications for diverse types of ordering fronts that occur under confinement conditions in biological or materials-processing contexts.

Forbidden interval of propagation speed for exothermic chemical fronts

We consider the propagation of an exothermic chemical front toward an unstable steady state. The hydrodynamic equations are solved numerically for increasing values of the activation energy of the reaction which controls the reaction front speed. For a large speed, the marginal stability criterion of the isothermal case is recovered. For a small speed, we observe two well-separated traveling waves: a heat front is preceding the reaction front. We find analytically a forbidden speed interval where the hydrodynamical system does not admit stationary traveling solutions.

Fisher waves and front roughening in a two-species invasion model with preemptive competition

We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits an effective power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of nonequilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of O(t(-2/3)).

Microscopic simulations of an exothermic chemical wave front: departure from the continuity equations

We perform microscopic simulations of a reactive dilute gas and study the propagation of an exothermic chemical wave front in an infinite, one-dimensional medium. The simulation results concerning front propagation speed and concentrations, temperature and stream velocity profiles are compared with the results of the integration of the macroscopic continuity equations in the case of fast reactions. The discrepancies between the two approaches are related to a chemically induced departure from local equilibrium.

Molecular-dynamics simulations and master-equation description of a chemical wave front: Effects of density and size of reaction zone on propagation speed

We compare the master-equation description and molecular-dynamics simulations of a chemical wave front. We find that the front propagation speed depends on the number of particles in the reaction zone. For the master equation the dependence follows the well-known logarithmic prediction obtained when introducing a cutoff into the macroscopic reaction-diffusion equation. The molecular-dynamics simulations reveal that the logarithmic law is compromised for dense fluids.

Population dynamics with global regulation: the conserved Fisher equation

We introduce and study a conserved version of the Fisher equation. Within a population biology context, this model describes spatially extended populations in which the total number of individuals is fixed due to either biotic or environmental factors. We find a rich spectrum of dynamical phases including a pseudotraveling wave and, in the presence of the Allee effect, a phase transition from a locally constrained high density state to a low density fragmented state.

Propagating waves in one-dimensional discrete networks of coupled units

We investigate the behavior of discrete systems on a one-dimensional lattice composed of localized units interacting with each other through nonlocal, nonlinear reactive dynamics. In the presence of second-order and third-order steps coupling two or three neighboring sites, respectively, we observe, for appropriate initial conditions, the propagation of waves which subsist in the absence of mass transfer by diffusion. For the case of the third-order (bistable) model, a counterintuitive effect is also observed, whereby the homogeneously less stable state invades the more stable one under certain conditions. In the limit of a continuous space the dynamics of these networks is described by a generic evolution equation, from which some analytical predictions can be extracted. The relevance of this mode of information transmission in spatially extended systems of interest in physical chemistry and biology is discussed.

Emergence of pulled fronts in fermionic microscopic particle models

We study the emergence and dynamics of pulled fronts described by the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation in the microscopic reaction-diffusion process A+A<-->A on the lattice when only a particle is allowed per site. To this end we identify the parameter that controls the strength of internal fluctuations in this model, namely, the number of particles per correlated volume. When internal fluctuations are suppressed, we explictly see the matching between the deterministic FKPP description and the microscopic particle model.

Crossover effects in a discrete deposition model with Kardar-Parisi-Zhang scaling

We simulated a growth model in (1+1) dimensions in which particles are aggregated according to the rules of ballistic deposition with probability p or according to the rules of random deposition with surface relaxation (Family model) with probability 1-p. For any p>0, this system is in the Kardar-Parisi-Zhang (KPZ) universality class, but it presents a slow crossover from the Edwards-Wilkinson class (EW) for small p. From the scaling of the growth velocity, the parameter p is connected to the coefficient lambda of the nonlinear term of the KPZ equation, giving lambda approximately p(gamma), with gamma=2.1+/-0.2. Our numerical results confirm the interface width scaling in the growth regime as W approximately lambda(beta)t(beta) and the scaling of the saturation time as tau approximately lambda(-1)L(z), with the expected exponents beta=1/3 and z=3/2, and strong corrections to scaling for small lambda. This picture is consistent with a crossover time from EW to KPZ growth in the form t(c) approximately lambda(-4) approximately p(-8), in agreement with scaling theories and renormalization group analysis. Some consequences of the slow crossover in this problem are discussed and may help investigations of more complex models.

Fluctuating pulled fronts: The origin and the effects of a finite particle cutoff

Recently, it has been shown that when an equation that allows the so-called pulled fronts in the mean-field limit is modeled with a stochastic model with a finite number N of particles per correlation volume, the convergence to the speed v(*) for N--> infinity is extremely slow-going only as ln(-2)N. Pulled fronts are fronts that propagate into an unstable state, and the asymptotic front speed is equal to the linear spreading speed v(*) of small linear perturbations about the unstable state. In this paper, we study the front propagation in a simple stochastic lattice model. A detailed analysis of the microscopic picture of the front dynamics shows that for the description of the far tip of the front, one has to abandon the idea of a uniformly translating front solution. The lattice and finite particle effects lead to a "stop-and-go" type dynamics at the far tip of the front, while the average front behind it "crosses over" to a uniformly translating solution. In this formulation, the effect of stochasticity on the asymptotic front speed is coded in the probability distribution of the times required for the advancement of the "foremost bin." We derive expressions of these probability distributions by matching the solution of the far tip with the uniformly translating solution behind. This matching includes various correlation effects in a mean-field type approximation. Our results for the probability distributions compare well to the results of stochastic numerical simulations. This approach also allows us to deal with much smaller values of N than it is required to have the ln(-2)N asymptotics to be valid. Furthermore, we show that if one insists on using a uniformly translating solution for the entire front ignoring its breakdown at the far tip, then one can obtain a simple expression for the corrections to the front speed for finite values of N, in which various subdominant contributions have a clear physical interpretation.

Kinematic reduction of reaction-diffusion fronts with multiplicative noise: derivation of stochastic sharp-interface equations

We study the dynamics of generic reaction-diffusion fronts, including pulses and chemical waves, in the presence of multiplicative noise. We discuss the connection between the reaction-diffusion Langevin-like field equations and the kinematic (eikonal) description in terms of a stochastic moving-boundary or sharp-interface approximation. We find that the effective noise is additive and we relate its strength to the noise parameters in the original field equations, to first order in noise strength, but including a partial resummation to all orders which captures the singular dependence on the microscopic cutoff associated with the spatial correlation of the noise. This dependence is essential for a quantitative and qualitative understanding of fluctuating fronts, affecting both scaling properties and nonuniversal quantities. Our results predict phenomena such as the shift of the transition point between the pushed and pulled regimes of front propagation, in terms of the noise parameters, and the corresponding transition to a non-Kardar-Parisi-Zhang universality class. We assess the quantitative validity of the results in several examples including equilibrium fluctuations and kinetic roughening. We also predict and observe a noise-induced pushed-pulled transition. The analytical predictions are successfully tested against rigorous results and show excellent agreement with numerical simulations of reaction-diffusion field equations with multiplicative noise.

Internal fluctuations effects on Fisher waves

We study the diffusion-limited reaction A+A<-->A in various spatial dimensions to observe the effect of internal fluctuations on the interface between stable and unstable phases. We find that, similar to what has been observed in d = 1 dimensions, internal fluctuations modify the mean-field predictions for this process, which is given by Fisher's reaction-diffusion equation. In d>1 the front displays local fluctuations perpendicular to the direction of motion which, with a proper definition of the interface, can be fully described within the Kardar-Parisi-Zhang (KPZ) universality class. This clarifies the apparent discrepancies with KPZ predictions reported recently.

Generalized empty-interval method applied to a class of one-dimensional stochastic models

In this work we study, on a finite and periodic lattice, a class of one-dimensional (bimolecular and single-species) reaction-diffusion models that cannot be mapped onto free-fermion models. We extend the conventional empty-interval method, also called interparticle distribution function (IPDF) method, by introducing a string function, which is simply related to relevant physical quantities. As an illustration, we specifically consider a model that cannot be solved directly by the conventional IPDF method and that can be viewed as a generalization of the voter model and/or as an epidemic model. We also consider the reversible diffusion-coagulation model with input of particles and determine other reaction-diffusion models that can be mapped onto the latter via suitable similarity transformations. Finally we study the problem of the propagation of a wave front from an inhomogeneous initial configuration and note that the mean-field scenario predicted by Fisher's equation is not valid for the one-dimensional (microscopic) models under consideration.

Slow crossover to Kardar-Parisi-Zhang scaling

The Kardar-Parisi-Zhang (KPZ) equation is accepted as a generic description of interfacial growth. In several recent studies, however, values of the roughness exponent alpha have been reported that are significantly less than that associated with the KPZ equation. A feature common to these studies is the presence of holes (bubbles and overhangs) in the bulk and an interface that is smeared out. We study a model of this type in which the density of the bulk and sharpness of the interface can be adjusted by a single parameter. Through theoretical considerations and the study of a simplified model we determine that the presence of holes does not affect the asymptotic KPZ scaling. Moreover, based on our numerics, we propose a simple form for the crossover to the KPZ regime.

Directed particle diffusion under "burnt bridges" conditions

We study random walks on a one-dimensional lattice that contains weak connections, so-called "bridges." Each time the walker crosses the bridge from the left or attempts to cross it from the right, the bridge may be destroyed with probability p; this restricts the particle's motion and directs it. Our model, which incorporates asymmetric aspects in an otherwise symmetric hopping mechanism, is very akin to "Brownian ratchets" and to front propagation in autocatalytic A+B-->2A reactions. The analysis of the model and Monte Carlo simulations show that for large p the velocity of the directed motion is extremely sensitive to the distribution of bridges, whereas for small p the velocity can be understood based on a mean-field analysis. The single-particle model advanced by us here allows an almost quantitative understanding of the front's position in the A+B-->2A many-particle reaction.

Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence

A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0< or =x< or =L-y< or =L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.

Universality class of fluctuating pulled fronts

It has recently been proposed that fluctuating "pulled" fronts propagating into an unstable state should not be in the standard Kardar-Parisi-Zhang (KPZ) universality class for rough interface growth. We introduce an effective field equation for this class of problems, and show on the basis of it that noisy pulled fronts in d+1 bulk dimensions should be in the universality class of the ((d+1)+1)D KPZ equation rather than of the (d+1)D KPZ equation. Our scenario ties together a number of heretofore unexplained observations in the literature, and is supported by previous numerical results.

Fluctuation effects in an epidemic model

We study a discrete epidemic model A+B-->2A in one and two dimensions (1D and 2D). In 1D for low concentration theta, we find that a depletion zone exists ahead of the front and the average velocity of the front approaches v=theta/2. In the 1D high concentration limit, we find that the velocity approaches v=1-e(-theta/2). In 2D, for low concentration we also find a depletion zone, and the velocity scales as v approximately theta(0.6), which is different from the scaling expected from the mean field approximation, v approximately theta(0.5). Analysis of the interface width scaling properties demonstrated that the front dynamics of this reaction are not governed by the Kardar-Parisi-Zhang equation.

Coarsening of a class of driven striped structures

The coarsening process in a class of driven systems exhibiting striped structures is studied. The dynamics is governed by the motion of the driven interfaces between the stripes. When two interfaces meet they coalesce thus giving rise to a coarsening process in which l(t), the average width of a stripe, grows with time. This is a generalization of the reaction-diffusion process A+A-->A to the case of extended coalescing objects, namely, the interfaces. Scaling arguments which relate the coarsening process to the evolution of a single driven interface are given, yielding growth laws for l(t), for both short and long times. We introduce a simple microscopic model for this process. Numerical simulations of the model confirm the scaling picture and growth laws. The results are compared to the case where the stripes are not driven and different growth laws arise.

Fluctuation and relaxation properties of pulled fronts: A scenario for nonstandard kardar-parisi-zhang scaling

We argue that while fluctuating fronts propagating into an unstable state should be in the standard Kardar-Parisi-Zhang (KPZ) universality class when they are pushed, they should not when they are pulled: The 1/t velocity relaxation of deterministic pulled fronts makes it unlikely that the KPZ equation is their proper effective long-wavelength low-frequency theory. Simulations in 2D confirm the proposed scenario, and yield exponents beta approximately 0.29+/-0.01, zeta approximately 0.40+/-0.02 for fluctuating pulled fronts, instead of the (1+1)D KPZ values beta = 1/3, zeta = 1/2. Our value of beta is consistent with an earlier result of Riordan et al., and with a recent conjecture that the exponents are the (2+1)D KPZ values.

Front propagation in one-dimensional autocatalytic reactions: the breakdown of the classical picture at small particle concentrations

The autocatalytic scheme A+B-->2A in a discrete particle system is studied in one dimension via Monte Carlo simulations. We find considerable differences in the results for the front velocities and front forms compared to the classical, continuous picture, which is only valid in the limit of very small reaction probabilities p. Interestingly, we also obtain front propagation velocities fairly below the classical minimal velocity.