2020/01/13 by Samuel Nordmann, Benoı̂t Perthame, Nordmann, Samuel +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.2001.04323
openalex publication_date 2020/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study an equation structured by age and a phenotypic trait describing the\ngrowth process of a population subject to aging, competition between\nindividuals, and mutations. This leads to a renewal equation which occurs in\nmany evolutionary biology problems. We aim to describe precisely the\nasymp-totic behavior of the solution, to infer properties that illustrate the\nconcentration and adaptive dynamics of such a population. This work is a\ncontinuation of [38] where the case without mutations is considered. When\nmutations are taken into account, it is necessary to control the corrector\nwhich is the main novelty of the present paper. Our approach consists in\ndefining, by the Hopf transform, a Hamilton-Jacobi equation with an effective\nHamiltonian as in homogenization problems. Its solution carries the singular\npart of the limiting density (typically Dirac masses) and the corrector defines\nthe weights. The main new result of this paper is to prove that the corrector\nis uniformly bounded, using only the global Lipschitz and semi-convexity\nestimates for the viscosity solution of the Hamilton-Jacobi equation. We also\nestablish the limiting equation satisfied by the corrector. To the best of our\nknowledge, this is the first example where such bounds can be proved in such a\ncontext.\n