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

Abstract

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction–diffusion equations u t ( t , x ) = Δ u ( t , x ) − u ( t , x ) + g ( u ( t − h , x ) ) ( ∗ ) , when g ∈ C 2 ( R + , R + ) has exactly two fixed points: x 1 = 0 and x 2 = 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.