一个对流选择方程的长时间行为
Long-time behaviour of an advection-selection equation
摘要：
本文研究了对流选择方程$$\partial_tn(t,x)+\nabla \cdot \left(f(x)n(t,x)\right)=\left(r(x)-ρ(t)\right)n(t,x),\quad ρ(t)=\int_{\mathbb{R}^d}{n(t,x)dx}\quad t\geq 0, \; x\in \mathbb{R}^d,$$的长时间行为，初始条件为$n(0, \cdot)=n^0$ .在适应动力学领域，这个方程通常描述了一个表型结构种群随时间的演化。在这种情况下，$x\mapsto n(t,x)$表示以表型性状$x$为特征的群体密度，平流项“$\nabla \cdot \left(f(x)n(t,x)\right)$”表示驱使个体朝向特定区域的细胞分化现象，选择项“$\left(r(x)-ρ(t)\right)n(t,x)$”表示群体的生长，其在整个群体大小$ρ(t)=\int_{\mathbb{R}^d}{n(t,x)dx}$中为逻辑类型。在一维情形$x\in \mathbb{R}$中，我们证明了该方程的解既可以收敛到加权Dirac质量，也可以收敛到$L^1$中的函数.根据参数$n^0$，$f$和$r$，我们确定这两个制度的收敛发生，我们指定的重量和点的狄拉克质量的支持，或表达的$L^1$ -功能达到。
英文摘要：
We study the long-time behaviour of the advection-selection equation $$\partial_tn(t,x)+\nabla \cdot \left(f(x)n(t,x)\right)=\left(r(x)-ρ(t)\right)n(t,x),\quad ρ(t)=\int_{\mathbb{R}^d}{n(t,x)dx}\quad t\geq 0, \; x\in \mathbb{R}^d,$$ 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 `$\left(r(x)-ρ(t)\right)n(t,x)$' the growth of the population, which is of logistic type through the total population size $ρ(t)=\int_{\mathbb{R}^d}{n(t,x)dx}$. 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.
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