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May  2008, 20(2): 407-423. doi: 10.3934/dcds.2008.20.407

Admissible wavefront speeds for a single species reaction-diffusion equation with delay

Elena Trofimchuk 1, and  Sergei Trofimchuk 2,

1. 

Department of Mathematics II, National Technical University ‘KPI’, Kyiv, United States
2. 

Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  July 2006 Revised  May 2007 Published  November 2007

We consider equation $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x))$(*) , when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=\kappa>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(\kappa)) =$ [ c*, c * ] such that (*) has at least one (possibly, non-monotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here c*$ >0$ is finite and c * $ \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound c*is sharp so that there are not wavefronts moving with speed $c < $ c*. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations (*) can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. (*) includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with non-monotone nonlinearity.
Keywords: Time-delayed reaction-diffusion equation, heteroclinic solutions, single species population models., non-monotone positive travelling fronts.
Mathematics Subject Classification: 34E0.
Citation: Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407
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