### Abstract

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.

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

Pages (from-to) | 857-877 |

Number of pages | 21 |

Journal | Journal of Mathematical Biology |

Volume | 34 |

Issue number | 8 |

Publication status | Published - 1996 |

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### Keywords

- Reaction-diffusion equations
- Spatio-temporal delay
- Time-periodic diffusion coefficients

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*34*(8), 857-877.

**Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations.** / Gourley, S. A.; Britton, N. F.; Chaplain, M. A J; Byrne, H. M.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 34, no. 8, pp. 857-877.

}

TY - JOUR

T1 - Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

AU - Gourley, S. A.

AU - Britton, N. F.

AU - Chaplain, M. A J

AU - Byrne, H. M.

PY - 1996

Y1 - 1996

N2 - Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.

AB - Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.

KW - Reaction-diffusion equations

KW - Spatio-temporal delay

KW - Time-periodic diffusion coefficients

UR - http://www.scopus.com/inward/record.url?scp=0343599074&partnerID=8YFLogxK

M3 - Article

VL - 34

SP - 857

EP - 877

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 8

ER -