In this paper, we provide a complete description of the selected spreading speed of systems of reaction-diffusion equations with unilateral coupling and prove the existence of anomalous spreading speeds for systems with monostable nonlinearities. Our work extends known results for systems with linear and quadratic couplings, and Fisher-KPP type nonlinearities. Our proofs rely on the construction of appropriate sub-and super-solutions.

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: the optimal trait for survival depends linearly on the spatial variable. The survival or extinction depends on the sign of an underlying principal eigenvalue. We investigate the survival case when the initial data satisfies a so-called heavy tail condition in the space-trait plane. Under these assumptions, we show that the solution propagates in the favorable direction of survival by accelerating. We derive some precise estimates on the location of the level sets corresponding to the total population in the space variable, regardless of their traits. Our analysis also reveals that the orientation of the initial heavy tail is of crucial importance.

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location x, an individual must have a trait close to some optimal trait y opt (x). Our main focus is to understand the effect of a nonlinear environmental gradient.

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.