3 years ago

# Convergence to travelling waves in Fisher’s population genetics model with a non-Lipschitzian reaction term

Peter Takáč, Pavel Drábek

### 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(xt), $$(x,t)\in \mathbb {R}\times \mathbb {R}_+$$ . 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(xt) 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.

DOI: 10.1007/s00285-017-1103-z

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