Concentration in Lotka-Volterra parabolic equations: an asymptotic-preserving scheme - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2022

Concentration in Lotka-Volterra parabolic equations: an asymptotic-preserving scheme

Abstract

In this paper, we introduce and analyze an asymptotic-preserving scheme for Lotka-Volterra parabolic equations. It is a class of nonlinear and nonlocal stiff equations, which describes the evolution of a population structured with phenotypic trait. In a regime of long time and small mutations, the population concentrates at a set of dominant traits. The dynamics of this concentration is described by a constrained Hamilton-Jacobi equation, which is a system coupling a Hamilton-Jacobi equation with a Lagrange multiplier determined by a constraint. This coupling makes the equation nonlocal. Moreover, the constraint does not enjoy much regularity, since it can have jumps. The scheme we propose is convergent in all the regimes, and enjoys stability in the long time and small mutations limit. Moreover, we prove that the limiting scheme converges towards the viscosity solution of the constrained Hamilton-Jacobi equation, despite the lack of regularity of the constraint. The theoretical analysis of the schemes is illustrated and complemented with numerical simulations.
Fichier principal
Vignette du fichier
CalvezHivertYoldas.pdf (1.92 Mo) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03635830 , version 1 (08-04-2022)

Identifiers

Cite

Vincent Calvez, Hélène Hivert, Havva Yoldaş. Concentration in Lotka-Volterra parabolic equations: an asymptotic-preserving scheme. 2022. ⟨hal-03635830⟩
99 View
37 Download

Altmetric

Share

Gmail Facebook X LinkedIn More