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In this paper we study the existence of one-dimensional travelling wave solutions u(x, t) = φ(x - ct) for the non-linear degenerate (at u = 0) reaction-diffusion equation ut = [D(u)ux]x + g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c = 0, 2. The existence of a unique value c* > 0 of c for which φ(x - c* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c ≠ c*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. © Springer-Verlag 1997.

Original publication




Journal article


Journal of Mathematical Biology

Publication Date





713 - 728