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

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 ) ∈ 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 ∈ R ) of every solution u ( x , t ) of the Cauchy problem to a single travelling wave U ( x - c t + ζ ) as t → ∞ . The speed c and the travelling wave U are determined uniquely by f , whereas the shift ζ is determined by the initial data.