We consider the Fisher-KPP equation with a nonlocal saturation effect defined through an interaction kernel φ(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transformφ(ξ) is positive or if the length σ of the nonlocal interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Our second concern is the study of traveling waves. We prove that this equation admits traveling wave solutions that connect u = 0 to an unknown positive steady state u ∞ (x), for all speeds c ≥ c *. The traveling wave connects to the standard state u ∞ (x) ≡ 1 under the aforementioned conditions:φ(ξ) > 0 or σ is sufficiently small. However, the wave is not monotonic for σ large.

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type: (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 pathforms 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 reactiondiffusion equations such that for an arbitrary ε > 0, any family of propagation sets (S t ) t 0 has to satisfy S t ⊂ {x ∈ R N , |x| εt} for large t. In connection with spreading properties, we derive some new unique-* Corresponding author. 2147 ness 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.

This paper is concerned with the existence of pulsating traveling fronts for the equation:

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation∂ t u - Δ u = f (t, u), x ∈ R N, t ∈ R, where f = f (t, u) is a KPP monostable nonlinearity which depends in a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = μ (t) u (1 - u), with μ ∈ L ∞ (R) such that ess inf t ∈ R μ (t) > 0. We first prove the existence of generalized transition waves (recently defined in Berestycki and Hamel (2007) ) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem

This paper deals with the generalized principal eigenvalue of the parabolic oper-where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.

International audienceWe establish in this article spreading properties for the solutions of equations of the type ∂ t u − a(x)∂ xx u − q(x)∂ 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 ≤ w such that lim t→+∞ sup 0≤x≤wt |u(t, x)−1| = 0 for all w ∈ (0, w) and lim t→+∞ sup x≥wt |u(t, x)| = 0 for all w > 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)

International audienceThis paper is concerned with the periodic principal eigenvalue k λ (µ) associated with the operator − d 2 dx 2 − 2λ d dx − µ(x) − λ 2 , (1) where λ ∈ R and µ is continuous and periodic in x ∈ R. Our main result is that k λ (µ *) ≤ k λ (µ), where µ * is the Schwarz rearrangement of the function µ. 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 µ * is the Steiner symmetrization of µ. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases k λ and we compute the limit of k λ when the period of the coefficients goes to 0. Lastly, we prove that, in dimension 1, rearranging the diffusion term decreases k λ. These results rely on some new formula for the periodic principal eigenvalue of a nonsymmetric operator

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reactiontransport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub-and supersolutions and prove their weak stability in a weighted L 2 space. In case of an unbounded velocity set, we prove a superlinear spreading. It appears that the rate of spreading depends on the decay at infinity of the velocity distribution. In the case of a Gaussian distribution, we prove that the front spreads as t 3/2 .