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

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

2005-08-04

arxiv.org/abs/math/0508098v3

Mathematics

Dynamical Systems

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

Scientific paper

We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x)) (*)$, 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.

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