Non-monotone 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 in delayed reaction-diffusion equations $u_t(t,x)=\mathrm{\Delta }u(t,x)-u(t,x)+g(u(t-h,x))(\ast )$, when $g\in C^2(R_{+},R_{+})$ has exactly two fixed points: $x_1=0$ and $x_2=a>0$. 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 $h$ grows. 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 $(\ast )$. As an example, we consider the diffusive Nicholson’s blowflies equation.