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

Abstract

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.

Spatial heterogeneity alters the trade-off between growth and dispersal during a range expansion

“…Some recent works have proposed to take into account genetic adaptation in these spatio-temporal models, thanks to an additional variable, say y (interpreted as a phenotypic trait), a mutation term modeled with a Laplace diffusion operator, and a nonlocal selection term (Alfaro et al, 2017, 2013; Alfaro and Peltier, 2021; Peltier, 2020). These models describe adaptation along an environmental gradient, that is, a gradual change in various factors in space that determine the phenotypic traits that are favored by their growth rate R ( x, y ).…”

“…Here, each spatial position x is associated with a different optimal trait, i.e ., a trait which leads to a maximal growth rate. The value of this optimal trait may be proportional to the position (Alfaro et al, 2013; Peltier, 2020), may depend periodically on x (Alfaro and Peltier, 2021), or may change with time (Alfaro et al, 2017). Another important part of this literature has been interested in the case where the trait is the diffusion coefficient D (Benichou et al, 2012; Berestycki et al, 2015; Bouin and Calvez, 2014; Bouin et al, 2012), and mostly focused on the acceleration of the range expansion in this case, due to the selection of the individuals with enhanced dispersal abilities.…”