3 years ago

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

Peter Takáč, Pavel Drábek


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.

Publisher URL: https://link.springer.com/article/10.1007/s00285-017-1103-z

DOI: 10.1007/s00285-017-1103-z

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