We obtain a criterion for pulsating front speed-up by general periodic incompressible flows in two dimensions and in the presence of KPP nonlinearities. We achieve this by showing that the ratio of the minimal front speed and the effective diffusivity of the flow is bounded away from zero and infinity by constants independent of the flow. We also study speed-up of reaction-diffusion fronts by various examples of flows in two and three dimensions.
We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits $c^*(A)/A$ and $D(A)/A^2$ as $A\to\infty$, where $c^*(A)$ is the minimal front speed and $D(A)$ the effective diffusivity.
This paper is concerned with the existence of pulsating travelling fronts for a KPP reaction-diffusion equation posed in a multi-dimensional periodic medium. We provide an alternative proof of the classic existence result. Our proof relies largely on the construction of a wave profile under a moving frame, which avoids many technical difficulties in dealing with degenerate elliptic equations. Intriguingly, our analysis also yields that the profile of the front propagating along each rational direction in $\mathbb{S}^{N-1}$ is periodic in time.
In this paper we are interested in propagation phenomena for nonlocal reaction-diffusion equations of the type: $δ_tu = J \times u - u + f (x, u) t \in R^+, x \in R^N$, where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.
We consider a periodic reaction diffusion system which, because of competition between $u$ and $v$, does not enjoy the comparison principle. It also takes into account mutations, allowing $u$ to switch to $v$ and vice versa. Such a system serves as a model in evolutionary epidemiology where two types of pathogens compete in a heterogeneous environment while mutations can occur, thus allowing coexistence.We first discuss the existence of nontrivial positive steady states, using some bifurcation technics. Then, to sustain the possibility of invasion when nontrivial steady states exist, we constr…
The notion of traveling wave, which typically refers to some particular spatio-temporal con- nections between two stationary states (typically, entire solutions keeping the same profile's shape through time), is essential in the mathematical analysis of propagation phenomena. They provide insight on the underlying dynamics, and an accurate description of large time behavior of large classes of solutions, as we will see in this paper. For instance, in an homogeneous framework, it is well-known that, given a fast decaying initial datum (for instance, compactly supported), the solution of a KPP t…
We use a new method in the study of Fisher-KPP reaction-diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction-diffusion equations in several spatial dimensions. Our method is based on the construction of sub- and super-solutions to the non-linear PDE from solutions of its linearization at zero.
We study front propagation phenomena for a large class of nonlocal KPP-type reaction-diffusion equations in oscillatory environments, which model various forms of population growth with periodic dependence. The nonlocal diffusion is an anisotropic integro-differential operator of order $α\in (0,2)$.
The aim of this paper is to study the generalized Fisher-KPP equation with nonlocal diffusion. In specific we prove the existence of a critical speed so that traveling front type solutions exist up to this critical speed and non-existence of traveling fronts below this critical value. Moreover, we obtain uniqueness, up to translation, and decay estimates of these traveling fronts.
We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type rea…
This paper is concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts, but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich. In the paper, we describe the class of transition fronts and we study their qualitative dynamical properties. In particular, we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the …
We prove existence of and construct transition fronts for a class of reaction- diffusion equations with spatially inhomogeneous Fisher-KPP type reactions and non-local diffusion. Our approach is based on finding these solutions as perturbations of appropriate solutions to the linearization of the PDE at zero. Our work extends a method introduced by one of us to study such questions in the case of classical diffusion.
We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.
We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We …
We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the so-called minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under …
We consider solutions of the KPP-type equations with a periodically varying reaction rate, and compactly supported initial data. It has been shown by M. Bramson in the case of the constant reaction rate that the lag between the position of such solutions and that of the traveling waves grows as (3/2) log(t). We generalize this result to the periodic case
We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.
We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is perf…
We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.
In this paper, we study gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flatt…
We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length. To prove the variational principle, we use…
This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When o…
The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly e…
We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for each $c<2\sqrt{f'(0)}\,$,} \] and we consider here those satisfying \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for some $c>2\sqrt{f'(0)}\,$.} \] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.
Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitud…
This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $$u_t =\nabla\cdot(A(z)\nabla u) +q(z)\cdot\nabla u+ f(z,u), t \in \mathbb{R}, z \in Ω,$$ propagating with a speed $c.$ In the case of a "combustion" nonlinearity, the speed $c$ exists and it is unique, while the front $u$ is unique up to a translation in $t.$ We give a $\min-\max$ and a $\max-\min$ formula for this speed $c.$ On the other hand, in the case of a "ZFK" or a "KPP" nonlinearity, there exists a…
We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable $x/t$ and in which the time and space derivatives are coupled together. We will first establish uniqueness results for …
In a recent paper by Cantrell, Cosner and Yu, two-component KPP systems with competition of Lotka--Volterra type were analyzed and their long-time behavior largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a …
We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling …
This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptoti…
We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^β)$ for some explicit $β\in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Diri…
We study the one-dimensional Fisher-KPP equation, with an initial condition $u_0(x)$ that coincides with the step function except on a compact set. A well-known result of M. Bramson states that, as $t\to+\infty$, the solution converges to a traveling wave located at the position $X(t)=2t-(3/2)\log t+x_0+o(1)$, with the shift $x_0$ that depends on $u_0$. U. Ebert and W. Van Saarloos have formally derived a correction to the Bramson shift, arguing that $X(t)=2t-(3/2)\log t+x_0-3\sqrtπ/\sqrt{t}+O(1/t)$. Here, we prove that this result does hold, with an error term of the size $O(1/t^{1-γ})$, for …
We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are two-fold. First, we prove the existence of a two-parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co-periodic perturbations. Secondly, we restrict to a specific class of exponential ke…
We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property…
In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\R, \eqno(1) $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\R, \eqno(2) $$ where $a_0(t)$ is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of posit…
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $ϕ$ at a position $\tilde σ(t) = 2t - (3/2)\log t + α_0- 3\sqrtπ/\sqrt{t}$, with error $O(t^{γ-1})$ for any $γ>0$. With their methods, we find a refined shift $σ(t) = \tilde σ(t) + μ_* (\log t)/t + α_1/t$ such that in the frame moving with $σ$, the solution $u$ satisfies $u(t,x) = ϕ(x) + ψ(x)/t + O(t^{γ-3/2})$ for a certain profile $ψ$ independent of initial data. The coefficient $α_1$ depends on initial data, bu…
This paper is devoted to the study of the asymptotic behaviors of the minimal speed of propagation of pulsating traveling fronts solving the Fisher-KPP reaction-advection-diffusion equation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. We consider a periodic heterogenous framework and we use the formula of Berestycki, Hamel and Nadirashvili \cite{bhn2} for the minimal speed of propagation to prove the asymptotics in any space dimension $N.$ We express the limits as the maxima of certain variational quantities over the…
We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve thi…
We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp $t…
We study the front of the solution to the F-KPP equation with randomized non-linearity. Under suitable assumptions on the randomness involving spatial mixing behavior and boundedness, we show that the front of the solution lags at most logarithmically in time behind the front of the solution of the corresponding linearized equation, i.e. the parabolic Anderson model. This can be interpreted as a partial generalization of Bramson's findings for the homogeneous setting. Building on this result, we establish functional central limit theorems for the fronts of the solutions to both equations.
This paper is concerned with the existence of pulsating front solutions in space-periodic media for a bistable two-species competition--diffusion Lotka--Volterra system. Considering highly competitive systems, a simple "high frequency or small amplitudes" algebraic sufficient condition for the existence of pulsating fronts is stated. This condition is in fact sufficient to guarantee that all periodic coexistence states vanish and become unstable as the competition becomes large enough.
We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur and Holling-Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig-MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts wi…
We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed,…
This paper is devoted to reaction-diffusion equations with bistable nonlinearities depending periodically on time. These equations admit two linearly stable states. However, the reaction terms may not be bistable at every time. These may well be a periodic combination of standard bistable and monostable nonlinearities. We are interested in a particular class of solutions, namely pulsating fronts. We prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts. …
The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. It it shown that these transition fronts have exact decaying rates as the space variable tends to infinity. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work (\cite{ShSh14-4}), these transition fronts are continuously di…
We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$. The level sets of solutions with step-like initial conditions are located at position $c_*t-\frac{3}{2λ_*}\ln t+\mathcal{O}(1)$ as $t\rightarrow+\infty$ for some explicit positive constants $c_*$ and $λ_*$. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed $c_*…
We consider a reactive Boussinesq system with no stress boundary conditions in a periodic domain which is unbounded in one direction. Specifically, we couple the reaction-advection-diffusion equation for the temperature, $T$, and the linearized Navier-Stokes equation with the Boussinesq approximation for the fluid flow, $u$. We show that this system admits smooth pulsating front solutions that propagate with a positive, fixed speed.
Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bista…
This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations with Kolmogrov-Petrovsky-Piskunov (KPP) type nonlinearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems ( proved in Berestycki, Hamel and Nadirashvili \cite{BHN1}), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviors and the variat…
