Asymptotic Wave Speed Research Articles

Asymptotic Behavior of Stochastic Reaction–Diffusion Equations

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can be obtained so that we can construct a sup-solution and a sub-solution to estimate the upper bound and the lower bound of wave speed.

Stochastic travelling wave of three-species competitive–cooperative systems with multiplicative noise

This paper is devoted to the stochastic travelling wave solutions of a three-species competitive–cooperative system with multiplicative noise. The existence of the travelling wave solutions can be proved by constructing upper/lower-solution and stochastic dynamical systems theory. Moreover, under the more restricted assumption on the coefficients of system, and applying Feynman–Kac formula and upper/lower solution technique, the upper/lower bounds of the asymptotic wave speed can be provided.

Travelling wave of stochastic Lotka-Volterra competitive system

<p style='text-indent:20px;'>This work is devoted to the travelling wave solutions and asymptotic wave speed for a two-species stochastic competitive system driven by a multiplicative Itô type white noise. By the introduction of sup-solutions and sub-solutions, the existence and the approximations of the travelling wave solutions are proven under some conditions on the parameters. Moreover, combining with the stochastic Feynman-Kac formula, sub-solution, sup-solution, monotone iteration and equilibria of the kinetic system corresponding to the transformed stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

Stochastic travelling wave solution of the $ N $-species cooperative systems with multiplicative noise

<abstract><p>The current paper is devoted to the stochastic $ N $-species cooperative system with a moderately strong noise. By the theory of monotone random systems and the technique of suitable marker of wavefront, the existence of the travelling wave solution is established. By applying the Feynman-Kac formula and sup-sub solution technique, the upper and lower bounded of the asymptotic wave speed are also obtained. Finally, we give an example for stochastic 3-species cooperative systems.</p></abstract>

Propagation of stochastic travelling waves of cooperative systems with noise

<p style='text-indent:20px;'>We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

Toxin-mediated competition in weakly motile bacteria

Many bacterial species produce toxins that inhibit their competitors. We model this phenomenon by extending classic two-species Lotka–Volterra competition in one spatial dimension to incorporate toxin production by one species. Considering solutions comprising two adjacent single-species colonies, we show how the toxin inhibits the susceptible species near the interface between the two colonies. Moreover, a sufficiently effective toxin inhibits the susceptible species to such a degree that an ‘inhibition zone’ is formed separating the two colonies. In the special case of truly non-motile bacteria, i.e. with zero bacterial diffusivity, we derive analytical expressions describing the bacterial distributions and size of the inhibition zone. In the more general case of weakly motile bacteria, i.e. small bacterial diffusivity, these two-colony solutions become travelling waves. We employ numerical methods to show that the wavespeed is dependent upon both interspecific competition and toxin strength; precisely which colony expands at the expense of the other depends upon the choice of parameter values. In particular, a sufficiently effective toxin allows the producer to expand at the expense of the susceptible, with a wavespeed magnitude that is bounded above as the toxin strength increases. This asymptotic wavespeed is independent of interspecific competition and due to the formation of the inhibition zone; when the colonies are thus separated, there is no longer direct competition between the two species and the producer can invade effectively unimpeded by its competitor. We note that the minimum toxin strength required to produce an inhibition zone increases rapidly with increasing bacterial diffusivity, suggesting that even moderately motile bacteria must produce very strong toxins if they are to benefit in this way.

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation: ∂tu=κ2uxx+u(1−u)dt+ϵu∂tζu0(x)=1(−∞,−1Nlog2)(x)+12e−Nx1[−1Nlog2,+∞)(x)where the stochastic differential is taken in the sense of Itô and ζ is a Gaussian random field satisfying Eζ=0 and Eζ(s,x)ζ(t,y)=(s∧t)Γ(x−y). Two situations are considered: firstly, ζ is simply a standard Wiener process (i.e. Γ≡1): secondly, Γ∈C∞(R) with lim|z|→+∞|Γ(z)|=0.The results are as follows: in the first situation (standard Wiener process: i.e. Γ(x)≡1), there is a non-degenerate travelling wave front if and only if ϵ22<1, with asymptotic wave speed max2κ(1−ϵ22),1N(1−ϵ22)+κN21{N<2κ(1−ϵ22)}; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on ϵ.In the second situation (noise with spatial covariance which decays to 0 at ±∞, stochastic integral taken in the sense of Itô), a travelling front can be defined for all ϵ>0. Its average asymptotic speed does not depend on ϵ and is the classical wave speed of the unperturbed KPP equation.

Gene drive through a landscape: Reaction–diffusion models of population suppression and elimination by a sex ratio distorter

Some genes or gene complexes are transmitted from parents to offspring at a greater-than-Mendelian rate, and can spread and persist in populations even if they cause some harm to the individuals carrying them. Such genes may be useful for controlling populations or species that are harmful. Driving-Y chromosomes may be particularly potent in this regard, as they produce a male-biased sex ratio that, if sufficiently extreme, can lead to population elimination. To better understand the potential of such genes to spread over a landscape, we have developed a series of reaction–diffusion models of a driving-Y chromosome in 1-D and radially-symmetric 2-D unbounded domains. The wild-type system at carrying capacity is found to be unstable to the introduction of driving-Y males for all models investigated. Numerical solutions exhibit travelling wave pulses and fronts, and analytical and semi-analytical solutions for the asymptotic wave speed under bounded initial conditions are derived. The driving-Y male invades the wild-type equilibrium state at the front of the wave and completely replaces the wild-type males, leaving behind, at the tail of the wave, a reduced- or zero-population state of females and driving-Y males only. In our simplest model of a population with one life stage and density-dependent mortality, wave speed depends on the strength of drive and the diffusion rate of Y-drive males, and is independent of the population dynamic consequences (suppression or elimination). Incorporating an immobile juvenile stage of fixed duration into the model reduces wave speed approximately in proportion to the relative time spent as a juvenile. If females mate just once in their life, storing sperm for subsequent reproduction, then wave speed depends on the movement of mated females as well as Y-drive males, and may be faster or slower than in the multiple-mating model, depending on the relative duration of juvenile and adult life stages. Numerical solutions are shown for parameter values that may in part be representative for Anopheles gambiae, the primary vector of malaria in sub-Saharan Africa.

Asymptotic wave speed estimation in weakly undulated ducts carrying mean flow

An analytical asymptotic solution for evaluating phase speed of acoustic waves transmitting through two-dimensional duct carrying mean flow and having periodically undulated walls is obtained. The study sheds light on the combined effect of mean flow and geometric periodicity on sound phase speed. Assumption of no fluid flow separation at the corrugated walls is made. The perturbation method of strained parameters is employed with two levels of approximation to provide an analytical expression for the acoustic phase speed. A quantitative assessment of the effect of undulation density, different undulation amplitudes at each wall of the duct, and the degree of wall undulations symmetry about the duct mid-plane are presented. The effect of forward mean flow on the acoustic phase speed in presence of wall undulations is compared to that of backward mean flow. It is found that accounting for the mean flow inside undulated ducts generates results that are significantly different from those obtained when the fluid is considered stationary.

Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition

On a one-dimensional string of cells, the juxtacrine signalling model for Delta–Notch lateral inhibition by Collier et al. [J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: A mathematical model of Delta–Notch intercellular interaction, J. Theoret. Biol. 183 (1996) 429–446] exhibits a predominant alternating pattern of cells expressing either Delta or Notch, as well as many aperiodic patterns. Despite this multistationarity, in the idealised situation of no noise, travelling waves invading the unstable, homogeneous state only generate the predominant alternating pattern in their wake all over the lattice. However, this robustness is totally lost in the presence of stochastic noise because the invaded, initial state is unstable. Using linear approximations around the initial, homogeneous state and around the final, patterned state, we are able to derive analytically all essential properties of the wave: the shape of the wave front, the unique, alternating pattern generated by the wave, and the asymptotic speed of the wave front. We show that the asymptotic wave speed equals the theoretical minimum wave speed. The latter agrees extremely well with the value estimated from numerical simulations. Thus, in this system travelling waves are pulled by the leading edge of the front.

Traveling waves in a model of influenza A drift

Between major pandemics, the influenza A virus changes its antigenic properties by accumulating point mutations (drift) mainly in the RNA genes that code for the surface proteins hemagglutinin (HA) and neuraminidase (NA). The successful strain (variant) that will cause the next epidemic is selected from a reduced number of progenies that possess relatively high transmissibility and the ability to escape from the immune surveillance of the host. In this paper, we analyse a one-dimensional model of influenza A drift (Z. Angew. Math. Mech. 76 (2) (1996) 421) that generalizes the classical SIR model by including mutation as a diffusion process in a phenotype space of variants. The model exhibits traveling wave solutions with an asymptotic wave speed that matches well those obtained from numerical simulations. As exact solutions for these waves are not available, asymptotic estimates for the amplitudes of infected and recovered classes are provided through an exponential approximation based on the smallness of the diffusion constant. Through this approximation, we find simple scaling properties to several parameters of relevance to the epidemiology of the disease.

Invasion by Extremes: Population Spread with Variation in Dispersal and Reproduction

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.

Invasion by Extremes: Population Spread with Variation in Dispersal and Reproduction

For populations having dispersal described by fat‐tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat‐tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment‐generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment‐generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat‐tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest‐forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as $$( 1/T) ( \pi uR_{0}/2) ^{1/2}$$ m yr−1, where T is generation time, and u is a distance parameter (m2) of Clark et al.’s 2Dt model having shape parameter $$p=1$$. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life‐history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.

Reaction-diffusion travelling waves in the acidic nitrate-ferroin reaction

A two-variable model proposed for the acidic nitrate-ferroin reaction is considered in the reaction-diffusion context. An initial-value problem in which an amount of nitrate is introduced locally into ferroin at uniform concentration is treated both analytically and numerically. It is shown that the large time structure is a reaction-diffusion travelling wave of permanent form propagating with constant speed. This asymptotic wave speed is shown to be the minimum possible wave speed and the asymptotic approach to this value is estimated. Properties of the permanent-form travelling waves are derived and solutions valid for small and large values of a parameter β, involved in the kinetic mechanism, are obtained.

Fluctuation effects on wave propagation in a reaction-diffusion process

The reaction-diffusion process corresponding to the Fisher-Kolmogorov equation is studied by means of a discrete multivariate master equation. For travelling wave fronts the stability criterion necessary for the applicability of a system-size expansion is shown to be violated due to the existence of a zero mode of the first variational equation. This zero mode is connected to the translational invariance of the system. Performing stochastic simulations of the master equation in a wide range of parameters it is demonstrated that for finite size of the system (up to about 10 7 particles in the frontal region) a rather large fluctuation effect on the wave propagation speed results: in general, the asymptotic wave speed lies below the stable, minimal speed which is given by a theorem of Kolmogorov for the macroscopic equation. The wave front position exhibits a diffusion-type behaviour associated with translative fluctuations along the propagation direction.