We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.

Using three different notions of the generalized principal eigenvalue of linear second-order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discusse

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −a_ij(x)∂_ij u(x) − q_i(x)∂_i u(x) = f (x,u(x)), x ∈ 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

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