2017/11/29 by Nicolas Champagnat, Champagnat, Nicolas, Benoît Henry +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Evolution and Genetic Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1711.10732
openalex publication_date 2017/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work is devoted to the study of scaling limits in small mutations and\nlarge time of the solutions u^\ε of two deterministic models of\nphenotypic adaptation, where the parameter \ε > 0 scales the size of\nmutations. The first model is the so-called Lotka-Volterra parabolic PDE in R d\nwith an arbitrary number of resources and the second one is an adaptation of\nthe first model to a finite phenotype space. The solutions of such systems\ntypically concentrate as Dirac masses in the limit \ε \→ 0.\nOur main results are, in both cases, the representation of the limits of\n\ε log u^\ε as solutions of variational problems and regularity\nresults for these limits. The method mainly relies on Feynman-Kac type\nrepresentations of u \ε and Varadhan's Lemma. Our probabilistic\napproach applies to multi-resources situations not covered by standard\nanalytical methods and makes the link between variational limit problems and\nHamilton-Jacobi equations with irregular Hamiltonians that arise naturally from\nanalytical methods. The finite case presents substantial difficulties since the\nrate function of the associated large deviation principle has non-compact level\nsets. In that case, we are also able to obtain uniqueness of the solution of\nthe variational problem and of the associated differential problem which can be\ninterpreted as a Hamilton-Jacobi equation in finite state space.\n