Populations facing a nonlinear environmental gradient: steady states and pulsating fronts

We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. This population is facing an {\it 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 {\it nonlinear} environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with $y_{opt}(x) = \varepsilon\theta(x)$, $0<\vert \varepsilon \vert \ll 1$. We construct steady states solutions and, when $\theta$ 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 $y$ we take advantage of a Hilbert basis of $L^{2}(\mathbb{R})$ made of eigenfunctions of an underlying Schr\"odinger operator, whereas to deal with the space variable $x$ 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.