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

Convergence to travelling waves in Fisher’s population genetics model with a non-Lipschitzian... 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), $$(x,t)\in \mathbb {R}\times \mathbb {R}_+$$ ( x , t ) ∈ R × 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 $$x\in \mathbb {R}$$ x ∈ R ) of every solution u(x, t) of the Cauchy problem to a single travelling wave $$U(x-ct + \zeta )$$ U ( x - c t + ζ ) as $$t\rightarrow \infty$$ t → ∞ . The speed c and the travelling wave U are determined uniquely by f, whereas the shift $$\zeta$$ ζ is determined by the initial data. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Biology Springer Journals

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

, Volume 75 (4) – Feb 14, 2017
44 pages

/lp/springer_journal/convergence-to-travelling-waves-in-fisher-s-population-genetics-model-7utGHWMvjK
Publisher
Springer Journals
Subject
Mathematics; Mathematical and Computational Biology; Applications of Mathematics
ISSN
0303-6812
eISSN
1432-1416
D.O.I.
10.1007/s00285-017-1103-z
Publisher site
See Article on Publisher Site

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

### Journal

Journal of Mathematical BiologySpringer Journals

Published: Feb 14, 2017

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