A Hamilton–Jacobi approach for a model of population structured by space and trait

Abstract

We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x ∈ R d and a phenotypical trait θ ∈ Θ. Considering diffusion, mutations and space-local competition between the individuals, we analyze the asymptotic (longtime/long-range in the x variable) exponential behavior of the solutions. Using some kind of real phase WKB ansatz, we prove that the propagation of the population in space can be described by a Hamilton-Jacobi equation with obstacle which is independent of θ. The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term.

Asymptotic analysis of a quantitative genetics model with nonlinear integral operator

“…We are interested in the asymptotic behaviour of the trait distribution F ε as ε vanishes. This asymptotic regime was investigated thoroughly for various linear operators B ε associated with asexual reproduction such as for instance the diffusion operator F ε (z)+ε 2 ∆F ε (z), or the convolution operator 1 ε K( z ε ) * F ε (z) where K is a probability kernel with unit variance, see Diekmann et al (2005); Perthame (2007); Barles and Perthame (2007); Barles et al (2009) ;Lorz et al (2011) for the earliest investigations, see further Méléard and Mirrahimi (2015); Mirrahimi (2018); Bouin et al (2018b) for the case of a fractional diffusion operator (or similarly a fat-tailed kernel K), and see further Mirrahimi (2013); Mirrahimi and Perthame (2015); Bouin and Mirrahimi (2015); Lam and Lou (2017); Gandon and Mirrahimi (2017); Mirrahimi (2017); Mirrahimi and Gandon (2018); for the interplay between evolutionary dynamics and a spatial structure. In the linear case, the asymptotic analysis usually leads to a Hamilton-Jacobi equation for the Hopf-Cole transform U ε = −ε log F ε .…”

Super-linear spreading in local and non-local cane toads equations

“…Since the linearizations of (1.6) and (1.7) are identical, we expect both models to have the same propagation speed. Models involving non-local reaction terms have been the subject of intense study in recent years due to the complexity of the dynamics -see, for example, [2,11,17,25,30,31] and references therein. The cane toads equation has similarly attracted recent interest, mostly when the motility set Θ is a finite interval.…”

Influence of a mortality trade-off on the spreading rate of cane toads fronts

“…To answer these questions, we focus on a cane-toad equation with mortality trade-off. This is a non-local reaction-diffusion-mutation equation that is a refinement of the now standard cane-toad equation proposed in [5] and investigated in [7,9,12,13,34] (see also [3,14,15,25,30] for a similar studies). We now introduce this model.…”