Wavefronts of a stage structured model with state--dependent delay

Abstract

This paper deals with a diffusive stage structured model withstate-dependent delay which is assumed to be an increasing functionof the population density. Compared with the constant delay, thestate--dependent delay makes the dynamic behavior more complex. Forthe state--dependent delay system, the dynamic behavior is dependentof the diffusion coefficients, while the equilibrium state ofconstant delay system is not destabilized by diffusion. Throughcalculating the minimum wave speed, we find that the wave is sloweddown by the state-dependent delay. Then, the existence of travelingwaves is obtained by constructing a pair of upper--lower solutionsand using Schauder's fixed point theorem. Finally, the travelingwavefront solutions for large wave speed are also discussed, and thefronts appear to be all monotone, regardless of the state dependentdelay. This is an interesting property, since many findings arefrequently reported that delay causes a loss of monotonicity, withthe front developing a prominent hump in some other delay models.