Non-monotone travelling waves in a single species reaction-diffusion equation with delay

Teresa Faria; Sergei Trofimchuk

doi:10.1016/j.jde.2006.05.006

paper

Non-monotone travelling waves in a single species reaction-diffusion equation with delay

arXiv:math/0508098 · doi:10.1016/j.jde.2006.05.006

Abstract

We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_{t} (t, x) = Δ u (t, x) - u (t, x) + g (u (t - h, x)) (*)$ , 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 $(*)$ . As an example, we consider the diffusive Nicholson's blowflies equation.

23 pages, several important modifications are made. Some references and comments added to the previous version. To appear in the Journal of Differential Equations

Non-monotone travelling waves in a single species reaction-diffusion equation with delay

Abstract

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