Long-time behaviour of an advection-selection equation (2301.02470v1)

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.