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

TL;DR

This paper develops and analyzes an asymptotic-preserving numerical scheme for Lotka-Volterra parabolic equations, effectively capturing population concentration dynamics and converging to the constrained Hamilton-Jacobi equation's viscosity solution.

Contribution

It introduces a novel scheme that remains stable and convergent across regimes and handles the nonlocal, irregular constraint in the limiting Hamilton-Jacobi equation.

Findings

Scheme converges in all regimes

Ensures stability in long-time, small mutation limit

Numerical simulations confirm theoretical results

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.