Spatial Noise Stabilizes Periodic Wave Patterns in Oscillatory Systems on Finite Domains

Abstract

Invasions in oscillatory systems generate in their wake spatiotemporal oscillations, consisting of either periodic wavetrains or irregular oscillations that appear to be spatiotemporal chaos. We have shown previously that when a finite domain, with zero-flux boundary conditions, has been fully invaded, the spatiotemporal oscillations persist in the irregular case, but die out in a systematic way for periodic traveling waves. In this paper, we consider the effect of environmental inhomogeneities on this persistence. We use numerical simulations of several predator-prey systems to study the effect of random spatial variation of the kinetic parameters on the die-out of regular oscillations and the long-time persistence of irregular oscillations. We find no effect on the latter, but remarkably, a moderate spatial variation in parameters leads to the persistence of regular oscillations, via the formation of target patterns. In order to study this target pattern production analytically, we turn to λ-ω systems. Numerical simulations confirm analagous behavior in this generic oscillatory system. We then repeat this numerical study using piecewise linear spatial variation of parameters, rather than random variation, which also gives formation of target patterns under certain circum- stances, which we discuss. We study this in detail by deriving an analytical approximation to the targets formed when the parameter λ0 varies in a simple, piecewise linear manner across the do- main, using perturbation theory. We end by discussing the applications of our results in ecology and chemistry.