How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. 
A classical result of Freidlin and Gartner  yields that the solutions of the initial value problems associated 
with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in 
 in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null 
average or rescaling the coefficients by the change of variables $x\to x/L$, with $L>1$, speeds up the propagation. From a modelling point of view, 
these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.

Related papers:

title: Asymptotic spreading in heterogeneous diffusive excitable media
abstract: We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction–diffusion equations of the type: ∂ t u − ∇ ⋅ ( A ( t , x ) ∇ u ) + q ( t , x ) ⋅ ∇ u = f ( t , x , u ) with compactly supported initial conditions at t = 0 . Here, A , q , f have a general dependence in t ∈ R + and x ∈ R N . We establish properties of families of propagation sets which are defined as families of subsets ( S t ) t ⩾ 0 of R N such that lim inf t → + ∞ { inf x ∈ S t u ( t , x ) } > 0 . The aim is to characterize such families as sharply as possible. In particular, we give some conditions under which: (1) a given path ( { ξ ( t ) } ) t ⩾ 0 , where ξ ( t ) ∈ R N , forms a family of propagation sets, or (2) one can find such a family with S t ⊃ { x ∈ R N , | x | ⩽ r ( t ) } and lim t → + ∞ r ( t ) = + ∞ . This second property is called here complete spreading. Furthermore, in the case q ≡ 0 and inf ( t , x ) ∈ R + × R N f u ′ ( t , x , 0 ) > 0 , as well as under some more general assumptions, we show that there is a positive spreading speed, that is, r ( t ) can be chosen so that lim inf t → + ∞ r ( t ) / t > 0 . In the general case, we also show the existence of an explicit upper bound C > 0 such that lim sup t → + ∞ r ( t ) / t C . On the other hand, we provide explicit examples of reaction–diffusion equations such that for an arbitrary e > 0 , any family of propagation sets ( S t ) t ⩾ 0 has to satisfy S t ⊂ { x ∈ R N , | x | ⩽ e t } for large t. In connection with spreading properties, we derive some new uniqueness results for the entire solutions of this type of equations. Lastly, in the case of space–time periodic media, we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero.

title: Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts
abstract: This paper is concerned with propagation phenomena for reaction–diffusion equations of the type: u t − ∇ ⋅ ( A ( x ) ∇ u ) = f ( x , u ) , x ∈ R N , where A is a given periodic diffusion matrix field, and f is a given nonlinearity which is periodic in the x-variables. This article is the sequel to [H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: I—influence of periodic heterogeneous environment on species persistence, ]. The existence of pulsating fronts describing the biological invasion of the uniform 0 state by a heterogeneous state is proved here. A variational characterization of the minimal speed of such pulsating fronts is proved and the dependency of this speed on the heterogeneity of the medium is also analyzed.

title: The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow
abstract: In this paper, we prove via counterexamples that adding an advection term of the form Shear flow (whose streamlines are parallel to the direction of propagation) to a reaction-diffusion equation will be an enough heterogeneity to spoil the increasing behavior of the KPP speed of propagation with respect to diffusion. The non-monotonicity of the speed with respect to diffusion will occur even when the reaction term and the diffusion matrices are considered homogeneous (do not depend on space variables). For the sake of completeness, we announce our results in a setting which allows domains with periodic perforations that may or may not be equal to the whole space $\mathbb{R}^N.$

title: Functional Integration And Partial Differential Equations
abstract: This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

title: A viscosity solution method for the spreading speed formula in slowly varying media
abstract: In this paper, we consider reaction-diffusion-advection equations in slowly periodi-cally oscillating media. We prove the existence of and we give explicit expressions of the asymptotic spreading speeds of invasion of the unstable state 0 in any direction, when the period of the invaded medium becomes infinitely large. The limiting sprea-ding speeds involve families of 1-periodic Hamilton-Jacobi equations. In the case of one-dimensional reaction-diffusion equations, we analyze the relative effects of small perturbations of the diffusion and the reaction coefficients, and we compare the spreading speeds in slowly oscillating media to the homogenized spreading speeds in rapidly oscillating media.

title: Some dependence results between the spreading speed and the coefficients of the space–time periodic Fisher–KPP equation
abstract: We investigate in this paper the dependence relation between the space–time periodic coefficients A, q and μ of the reaction–diffusion equation \begin{linenomath}\[ \partial_t u -\nabla \cdot (A(t, x)\nabla u)+q(t, x)\cdot \nabla u = \mu(t, x)u(1-u), \] \end{linenomath} and the spreading speed of the solutions of the Cauchy problem associated with compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of μ decreases the minimal speed, (2) if μ is not constant with respect to x, then increasing the amplitude of the diffusion matrix A does not necessarily increase the minimal speed and (3) if A = IN, μ is a constant, then the introduction of a space periodic drift term q = ∇Q decreases the minimal speed. In order to prove these results, we use a variational characterisation of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearisation of the equation near zero. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients.

title: KPP pulsating front speed-up by flows
abstract: 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.

title: Traveling periodic waves in heterogeneous environments
abstract: A model for a single species population which propagates in a heterogeneous environment in a one dimensional space is presented. The environment is composed of two kinds of patches with different diffusivities and intrinsic growth rates, which are alternately arranged along the spatial axis. From the stability analysis of the model, the invasion condition for a new migrating species is obtained in terms of the sizes of patches, diffusivities, and growth rates. When the parameters satisfy the invasion condition, the distribution of the population initially localized in a bounded area always evolves into a traveling periodic wave (TPW) with a constant speed. When the invasion condition is not satisfied, the population fails in invasion tending to extinction. The velocity of TPW is calculated by using a dispersion relation, and the effects of the heterogeneity of the environment on the velocity are discussed.

title: Spreading speeds for one-dimensional monostable reaction-diffusion equations
abstract: We establish in this article spreading properties for the solutions of equations of the type $\partial$ t u -- a(x)$\partial$ xx u -- q(x)$\partial$ x u = f (x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable KPP type between two steady states 0 and 1 and the initial datum is compactly sup-ported. Using homogenization techniques, we construct two speeds w $\le$ w such that lim t$\rightarrow$+$\infty$ sup 0$\le$x$\le$wt |u(t, x)--1| = 0 for all w $\in$ (0, w) and lim t$\rightarrow$+$\infty$ sup x$\ge$wt |u(t, x)| = 0 for all w \textgreater{} w. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particu-lar, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w).

title: Stochastic homogenization of Hamilton–Jacobi equations and some applications
abstract: Homogenization-type results for the Cauchy problem for first-order PDE (Hamilton-Jacobi equations) are presented. The main assumption is that the Hamiltonian is superlinear and convex with respect to the gradient and stationary and ergodic with respect to the spatial variable. Some of applications to related problems as well as to the asymptotics of reaction-diffusion equations and turbulent combustion are also presented.

title: Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media
abstract: We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.

title: Traveling waves in a one-dimensional random medium
abstract: We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.

title: A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows
abstract: We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

title: The speed of propagation for KPP type problems. I: Periodic framework
abstract: This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article \cite{bh}. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.

title: Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena
abstract: This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.

title: Enhancement of the traveling front speeds in reaction-diffusion equations with advection
abstract: We establish rigorous lower bounds on the speed of traveling fronts and on the bulk burning rate in reaction-diffusion equation with passive advection. The non-linearity is assumed to be of either KPP or ignition type. We consider two main classes of flows. Percolating flows, which are characterized by the presence of long tubes of streamlines mixing hot and cold material, lead to strong speed-up of burning which is linear in the amplitude of the flow, U. On the other hand the cellular flows, which have closed streamlines, are shown to produce weaker increase in reaction. For such flows we get a lower bound which grows as U1/5 for a large amplitude of the flow.

title: The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator
abstract: This paper is concerned with the periodic principal eigenvalue $k_\lambda(\mu)$ associated with the operator $-\frac{d^2}{dx^2}-2\lambda\frac{d}{dx}-\mu(x)-\lambda^2$, where $\lambda\in\mathbb{R}$ and $\mu$ is continuous and periodic in $x\in\mathbb{R}$. Our main result is that $k_\lambda(\mu^*)\leq k_\lambda(\mu)$, where $\mu^*$ is the Schwarz rearrangement of the function $\mu$. From a population dynamics point of view, using reaction-diffusion modeling, this result means that the fragmentation of the habitat of an invading population slows down the invasion. We prove that this property does not hold in higher dimension if $\mu^*$ is the Steiner symmetrization of $\mu$. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases $k_\lambda$, and we compute the limit of $k_\lambda$ when the period of the coefficients goes to 0. Last, we prove that in dimension 1, rearranging the diffusion term decreases $k_\lambda$. These results rely on some new formula f...

title: Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations
abstract: We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities – including multistable ones – and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.

title: Maximizing the spreading speed of KPP fronts in two-dimensional stratified media
abstract: We consider the equation $u_t=u_{xx}+u_{yy}+b(x)f(u)+g(u)$, $(x,y)\in\mathbb R^2$ with monostable nonliearity, where $b(x)$ is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case where $b(x)$ is a smooth periodic function, there exists a pulsating travelling wave that propagates in the direction $(\cos\theta,\sin\theta)$ -- with average speed $c$ if and only if $c\geq c^*(\theta,b),$ where $c^*(\theta,b)$ is a certain positive number depending on $b.$ Moreover, the quantity $w(\theta;{b})=\min_{|\theta-\phi|<\frac{\pi}{2}}c^*(\phi;{b})/\cos(\theta-\phi)$ is called the spreading speed. This theory can be extended by showing the existence of the minimal speed $c^*(\theta,b)$ for any nonnegative measure $b$ with period $L.$ We then study the question of maximizing $c^*(\theta,b)$ under the constraint $\int_{[0,L)}b(x)dx=\alpha L,$ where $\alpha$ is an arbitrarily given positive constant. We prove that the maximum is attained by periodically arrayed Dirac's delta functions $h(x)=\alpha L\sum_{k\in\mathbb Z}\delta(x+kL)$ for any direction $\theta$. Based on these results, for the case that $b=h$ we also show the monotonicity of the spreading speedsin $\theta$ and study the asymptotic shape of spreading fronts for large $L$ and small $L$ . Finally, we show that for general 2-dimensional periodic equation $u_t=u_{xx}+u_{yy}+b(x,y)f(u)+g(u)$, $(x,y)\in\mathbb R^2$, the similar conclusions do not hold.

title: Liouville-type results for semilinear elliptic equations in unbounded domains
abstract: AbstractThis paper is devoted to the study of some class of semilinear elliptic equations in the whole space: 

$$ -a_{ij}(x)\partial_{ij}u(x) -- q_i(x)\partial_iu(x)=f(x,u(x)),\quad x\in{\mathbb R}^N. $$
 The aim is to prove uniqueness of positive- bounded solutions—Liouville-type theorems. Along the way, we establish also various existence results.We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation.