Long-time behaviour of an advection-selection equation

Abstract

We study the long-time behaviour of the advection-selection equation $${\partial }_{t}n(t,x)+\nabla \cdot \left(f(x)n(t,x)\right)=\left(r(x)-\rho (t)\right)n(t,x),\rho (t)={\int }_{{\mathbb{R}}^d}n(t,x)dxt\ge 0,x\in {\mathbb{R}}^d,$$ 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.