Nonmonotone travelling waves in a single species reaction–diffusion equation with delay

We prove the existence of a continuous family of positive and generally non-monotone travelling fronts for delayed reaction-diffusion equations ut(t, x) = �u(t, x)−u(t, x)+ g(u(t − h, x)) (∗), when g ∈ C 2 (R+, R+) has exactly two fixed points: x1 = 0 and x2 = K > 0. Recently, non-monotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass type equations with diffusion fall within this subclass of (∗). As an example, we consider the diffusive Nicholson’s blowflies equation.