Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts

TL;DR

This paper investigates the long-term oscillatory behavior of a population model governed by a linear growth-fragmentation equation, revealing convergence to an oscillatory state and introducing a numerical scheme to capture these dynamics.

Contribution

It demonstrates the convergence towards oscillatory solutions in a growth-fragmentation model and develops a numerical scheme capable of accurately capturing these oscillations.

Findings

Solution converges to an oscillatory function based on dominant eigenvectors.

The proof employs a relative entropy argument in a weighted $L^2$ space.

A non-dissipative numerical scheme effectively captures oscillations.

Abstract

We study the asymptotic behaviour of the following linear growth-fragmentation equation$$\dfrac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{\partial x} \big(x u(t,x)\big) + B(x) u(t,x) =4 B(2x)u(t,2x),$$ and prove that under fairly general assumptions on the division rate $B(x),$ its solution converges towards an oscillatory function,explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypo-coercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted $L^2$ space, where well-posedness is obtained via semigroup analysis. We also propose a non-dissipative numerical scheme, able to capture the oscillations.