Ricerc@Sapienza

We introduce a model designed to account for the influence of a line with fast diffusion–such as a road or another transport network–on the dynamics of a population in an ecological niche.This model consists of a system of coupled reaction-diffusion equations set on domains with different dimensions (line / plane). We first show that, in a stationary climate, the presence of the line is always deleterious and can even lead the population to extinction. Next, we consider the case where the niche is subject to a displacement, representing the effect of a climate change.

This paper develops the notion and properties of the generalized principal eigenvalue for an elliptic system coupling an equation in a plane with one on a line in this plane, together with boundary conditions that express exchanges taking place between the plane and the line. This study is motivated by the reaction-diffusion model introduced by Berestycki, Roquejoffre and Rossi [The influence of a line with fast diffusion on Fisher-KPP propagation, J. Math. Biol. 66(4-5) (2013) 743-766] to describe the effect on biological invasions of networks with fast diffusion imbedded in a field.

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.