论文标题
Lotka-Volterra抛物线方程中的浓度：渐近保护方案
Concentration in Lotka-Volterra parabolic equations: an asymptotic-preserving scheme
论文作者
论文摘要
在本文中，我们介绍和分析了Lotka-Volterra抛物线方程的渐近保护方案。它是一类非线性和非局部刚性方程，它描述了具有表型性状构成的种群的演变。在长期和小突变的状态下，人口集中在一组主要的特征上。这种浓度的动力学是通过受约束的汉密尔顿 - 雅各比方程来描述的，汉密尔顿 - 雅各布方程是一个系统，将汉密尔顿 - 雅各比方程与由约束确定的拉格朗日乘数耦合。这种耦合使方程式非本地。此外，由于可能会跳跃，因此约束并不享有太大的规律性。 我们提出的方案在所有制度中都是收敛的，并且长期以来稳定，并且较小的突变限制。此外，我们证明，尽管缺乏约束性的规律性，但限制方案仍会趋向于受约束的汉密尔顿 - 雅各比方程的粘度解。这些方案的理论分析被说明并与数值模拟相辅相成。
In this paper, we introduce and analyze an asymptotic-preserving scheme for Lotka-Volterra parabolic equations. It is a class of nonlinear and nonlocal stiff equations, which describes the evolution of a population structured with phenotypic trait. In a regime of long time and small mutations, the population concentrates at a set of dominant traits. The dynamics of this concentration is described by a constrained Hamilton-Jacobi equation, which is a system coupling a Hamilton-Jacobi equation with a Lagrange multiplier determined by a constraint. This coupling makes the equation nonlocal. Moreover, the constraint does not enjoy much regularity, since it can have jumps. The scheme we propose is convergent in all the regimes, and enjoys stability in the long time and small mutations limit. Moreover, we prove that the limiting scheme converges towards the viscosity solution of the constrained Hamilton-Jacobi equation, despite the lack of regularity of the constraint. The theoretical analysis of the schemes is illustrated and complemented with numerical simulations.
