Periodic traveling waves with large speed.

Periodic traveling waves (PTWs, wavetrains) have been extensively used to understand spatiotemporal oscillations observed in a number of field datasets. The aim of this paper is to investigate the existence, period functions and stability of PTWs with large speed in general two component reaction-diffusion systems. We first show the existence of small-amplitude (or large-amplitude) PTWs arising from Hopf bifurcation around a weak focus of order m (or from bifurcation of periodic orbits around a m-fold limit cycle). Second, we give the dominant terms of their period functions, which rarely appear in the existing references. Third, we establish the instability of PTWs in terms of the first-order derivative of their period functions. Our results indicate that the stability of small-amplitude PTWs depends on not only the real parts of the Lyapunov coefficients but also their imaginary parts. The invariant manifold theory developed by Fenichel is applied, along with bifurcation theory of planar dynamical systems. Finally, we apply the main results to a host-generalist parasitoid model for the biological control of a leafminer population. [ABSTRACT FROM AUTHOR]