title: Convergence to travelling waves in Fisher's population genetics model with a non-Lipschitzian reaction term.
abstract: We consider a one-dimensional population genetics model for the advance of an advantageous gene. The model is described by the semilinear Fisher equation with unbalanced bistable non-Lipschitzian nonlinearity f(u). The "nonsmoothness" of f allows for the appearance of travelling waves with a new, more realistic profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions u(x, t), [Formula: see text]. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave U. Our main result is the uniform convergence (for [Formula: see text]) of every solution u(x, t) of the Cauchy problem to a single travelling wave [Formula: see text] as [Formula: see text]. The speed c and the travelling wave U are determined uniquely by f, whereas the shift [Formula: see text] is determined by the initial data.
authors.affiliation: Department of Mathematics, N.T.I.S. (Center of New Technologies for Information Society), University of West Bohemia, P.O. Box 314, 306 14, Plzeň, Czech Republic.
authors.forename: Pavel
authors.initials: P
authors.lastname: Drábek
authors.affiliation: Institut für Mathematik, Universität Rostock, Ulmenstraße 69, Haus 3, 18055, Rostock, Germany. peter.takac@uni-rostock.de.
authors.forename: Peter
authors.identifier: http://orcid.org/0000-0002-8813-1122
authors.initials: P
authors.lastname: Takáč
mesh_terms: D000818:Animals; D005828:Genetics, Population / Q000706:statistics & numerical data*; D006801:Humans; D055641:Mathematical Concepts; D008957:Models, Genetic; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D011157:Population Dynamics / Q000706:statistics & numerical data
keywords: Fisher’s equation; Long-time convergence to a travelling wave; Mild solution of the Cauchy problem; Moving bounded habitat; Unbalanced bistable nonsmooth nonlinearity