Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay

We study positive bounded wave solutions u(t,x)= (ν x+ct), (−∞)=0, of equation ut(t,x)=Δu(t,x)−u(t,x)+g(u(t−h,x)), . This equation is assumed to have two non-negative equilibria: u1≡0 and u2≡κ>0. The birth function is unimodal and differentiable at 0 and κ. Some results also require the feedback condition (g(s)−κ)(s−κ)<0, with s [g(maxg),maxg] {κ}. If additionally (+∞)=κ, the above wave solution u(t,x) is called a travelling front. We prove that every wave (ν x+ct) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate such that Eq. (*) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c>c*. Our results are based on a detailed geometric description of the wave profile . In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass of ‘asymmetric’ tent maps such that given , there exists exactly one positive travelling front for each fixed admissible speed.