### 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 ?-? 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 circumstances, which we discuss. We study this in detail by deriving an analytical approximation to the targets formed when the parameter ?_{0} varies in a simple, piecewise linear manner across the domain, using perturbation theory. We end by discussing the applications of our results in ecology and chemistry. © 2000 Society for Industrial and Applied Mathematics.

Original language | English |
---|---|

Pages (from-to) | 1013-1041 |

Number of pages | 29 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 61 |

Issue number | 3 |

Publication status | Published - 2000 |

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### Keywords

- Oscillatory systems
- Periodic waves
- Spatial noise

### Cite this

*SIAM Journal on Applied Mathematics*,

*61*(3), 1013-1041.

}

*SIAM Journal on Applied Mathematics*, vol. 61, no. 3, pp. 1013-1041.

**Spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains.** / Kay, Alison L.; Sherratt, Jonathan A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains

AU - Kay, Alison L.

AU - Sherratt, Jonathan A.

PY - 2000

Y1 - 2000

N2 - 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 ?-? 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 circumstances, which we discuss. We study this in detail by deriving an analytical approximation to the targets formed when the parameter ?0 varies in a simple, piecewise linear manner across the domain, using perturbation theory. We end by discussing the applications of our results in ecology and chemistry. © 2000 Society for Industrial and Applied Mathematics.

AB - 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 ?-? 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 circumstances, which we discuss. We study this in detail by deriving an analytical approximation to the targets formed when the parameter ?0 varies in a simple, piecewise linear manner across the domain, using perturbation theory. We end by discussing the applications of our results in ecology and chemistry. © 2000 Society for Industrial and Applied Mathematics.

KW - Oscillatory systems

KW - Periodic waves

KW - Spatial noise

UR - http://www.scopus.com/inward/record.url?scp=0034912178&partnerID=8YFLogxK

M3 - Article

VL - 61

SP - 1013

EP - 1041

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -