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

We study positive bounded wave solutions u(t,x)=ϕ(ν⋅x+ct), ϕ(−∞)=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 g∈C(R+,R+) 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 ϕ(+∞)=κ, the above wave solution u(t,x) is called a travelling front. We prove that every wave ϕ(ν⋅x+ct) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c∗∈R+∪{+∞} 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 ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ‘asymmetric’ tent maps such that given g∈G, there exists exactly one positive travelling front for each fixed admissible speed.