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

AbstractWe prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction–diffusion equations ut(t,x)=Δu(t,x)−u(t,x)+g(u(t−h,x))(∗), when g∈C2(R+,R+) has exactly two fixed points: x1=0 and x2=K>0. Recently, nonmonotonic 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