Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as
models of biological phenomena. This paper begins with a survey of applications to
ecology, cell biology and bacterial colony patterns. The author then reviews mathematical
results on the existence of travelling wave front solutions of these equations, and their
generation from given initial data. A detailed study is then presented of the form of
smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the
particular equation ut =
(uux)x + u(1
− u). Using singular perturbation theory, the author derives an
asymptotic approximation to the wave, which gives valuable information about the structure
of smooth-front solutions. The approximation compares well with numerical results.