In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem:where Ω ⊂ R n is an open connected set, J a non-negative kernel and g a positive function. First, we establish a criterion for the existence of a principal eigenpair (λ p , φ p ). We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterise the solutions of some nonlinear nonlocal reaction diffusion equations.

We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits.

We study the travelling wave problem J * u - u - cu' + f(u) = 0 in R, u(-infinity) = 0, u(+infinity) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c not equal 0. For c = 0 we show examples of nonuniqueness

In this note, we give a positive answer to a question addressed in [8]. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are "rapid") of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with fat tails.

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c * > 0, and prove the existence of waves when c ≥ c * and the nonexistence when 0 ≤ c < c * .