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