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), $(x,t)\in \mathbb {R}\times \mathbb {R}_+$. 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 $x\in \mathbb {R}$) of every solution u(x, t) of the Cauchy problem to a single travelling wave $U(x-ct + \zeta )$ as $t\rightarrow \infty$. The speed c and the travelling wave U are determined uniquely by f, whereas the shift $\zeta$ is determined by the initial data.