Long-time behaviour of an advection-selection equation

TL;DR

This paper analyzes the long-term behavior of an advection-selection equation modeling phenotype evolution, showing conditions under which solutions converge to a Dirac mass or an integrable function, with explicit characterization.

Contribution

It provides a detailed classification of the asymptotic regimes for solutions of the advection-selection equation in one dimension, including explicit descriptions of the limits.

Findings

Solutions can converge to a Dirac mass or an L^1 function.

The regime depends on initial data, advection, and growth functions.

Explicit formulas for the limiting measures are derived.

Abstract

We study the long-time behaviour of the advection-selection equation $$\partial_tn(t,x)+\nabla \cdot \left(f(x)n(t,x)\right)=\left(r(x)-\rho(t)\right)n(t,x),\quad \rho(t)=\int_{\mathbb{R}^d}{n(t,x)dx}\quad t\geq 0, \; x\in \mathbb{R}^d,$$ with an initial condition $n(0, \cdot)=n^0$. In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, $x\mapsto n(t,x)$ represents the density of the population characterised by a phenotypic trait $x$, the advection term `$\nabla \cdot \left(f(x)n(t,x)\right)$' a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term `$\left(r(x)-\rho(t)\right)n(t,x)$' the growth of the population, which is of logistic type through the total population size $\rho(t)=\int_{\mathbb{R}^d}{n(t,x)dx}$. In the one-dimensional case $x\in \mathbb{R}$, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in $L^1$. Depending on the parameters $n^0$, $f$ and $r$, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the $L^1$-function which is reached.