论文标题
按年龄和具有突变的表型性状结构的人群中的浓度动力学。校正器的收敛性
Dynamics of concentration in a population structured by age and a phenotypic trait with mutations. Convergence of the corrector
论文作者
论文摘要
我们研究了按年龄和表型特征构成的方程式，该方程描述了人口的生长过程，受到老龄化，个人之间的竞争和突变。这导致了一个更新方程，该方程发生在许多进化生物学问题中。我们的目的是准确描述解决方案的Asymp-Toic行为，以推断出说明这种人群的浓度和适应性动态的特性。这项工作是[38]的延续，其中考虑了没有突变的情况。当考虑突变时，有必要控制校正器，这是本文的主要新颖性。我们的方法在于通过HOPF变换来定义具有有效汉密尔顿的汉密尔顿 - 雅各比方程，如均质问题。它的解决方案带有极限密度的单数部分（通常是狄拉克质量），校正器定义了权重。本文的主要新结果是证明校正器是统一的，仅使用全球Lipschitz和汉密尔顿 - 雅各比方程的粘度解的半偶像估计值。我们还建立了校正器满足的限制方程。据我们所知，这是可以在这种情况下证明这样的界限的第一个示例。
We study an equation structured by age and a phenotypic trait describing the growth process of a population subject to aging, competition between individuals, and mutations. This leads to a renewal equation which occurs in many evolutionary biology problems. We aim to describe precisely the asymp-totic behavior of the solution, to infer properties that illustrate the concentration and adaptive dynamics of such a population. This work is a continuation of [38] where the case without mutations is considered. When mutations are taken into account, it is necessary to control the corrector which is the main novelty of the present paper. Our approach consists in defining, by the Hopf transform, a Hamilton-Jacobi equation with an effective Hamiltonian as in homogenization problems. Its solution carries the singular part of the limiting density (typically Dirac masses) and the corrector defines the weights. The main new result of this paper is to prove that the corrector is uniformly bounded, using only the global Lipschitz and semi-convexity estimates for the viscosity solution of the Hamilton-Jacobi equation. We also establish the limiting equation satisfied by the corrector. To the best of our knowledge, this is the first example where such bounds can be proved in such a context.
