Analytic solutions for monotonic and oscillating fronts in a reaction–diffusion system under external fields

A piecewise linear reaction–diffusion system including an external field is considered. Analytic solutions are obtained for the propagating front, the front velocity, the perturbed front and the growth rate of perturbations. It is found that, when the ratio of the time scales, null-cline parameter and the external field are small enough, the moving front has an oscillating behaviour, similar to that of a front in an oscillatory medium. A comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases. In particular, we find that for fronts oscillating about a stable state both concentrations must oscillate in two-variable models, whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic.