### Abstract

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

Pages (from-to) | 155-173 |

Number of pages | 18 |

Journal | Applied Numerical Mathematics |

Volume | 18 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 1995 |

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### Keywords

- discretised evolution equation
- diffusion
- linear stability limit
- bifurcation patterns
- mathematics

### Cite this

*Applied Numerical Mathematics*,

*18*(1-3), 155-173. https://doi.org/10.1016/0168-9274(95)00051-U

}

*Applied Numerical Mathematics*, vol. 18, no. 1-3, pp. 155-173. https://doi.org/10.1016/0168-9274(95)00051-U

**Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay.** / Higham, D.J.; Sardar, T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay

AU - Higham, D.J.

AU - Sardar, T.

PY - 1995

Y1 - 1995

N2 - The long-time behaviour of a discretised evolution equation is studied. The equation, which involves diffusion and a nonlinear, delayed, reaction term, has been proposed as a model in population dynamics. It contains, as special cases, logistic-style problems that have been used before to provide canonical examples of spurious behaviour. The existence and stability of the basic steady states are systematically studied, as functions of the grid spacings and problem parameters. Particular attention is paid to the effect of the delay on the long-time behaviour. It is found that, as has been seen with other nonlinear problems, increasing the time step beyond the linear stability limit may induce stable, spurious, steady states, which are clearly undesirable as numerical solutions. When a delay is present, spurious solutions are also found to exist within the linear stability limit, and this is seen to affect the dynamics. Potential symmetry in the problem is identified and it is shown that in certain circumstances the bifurcation patterns depend dramatically upon whether the initial data shares the symmetry.

AB - The long-time behaviour of a discretised evolution equation is studied. The equation, which involves diffusion and a nonlinear, delayed, reaction term, has been proposed as a model in population dynamics. It contains, as special cases, logistic-style problems that have been used before to provide canonical examples of spurious behaviour. The existence and stability of the basic steady states are systematically studied, as functions of the grid spacings and problem parameters. Particular attention is paid to the effect of the delay on the long-time behaviour. It is found that, as has been seen with other nonlinear problems, increasing the time step beyond the linear stability limit may induce stable, spurious, steady states, which are clearly undesirable as numerical solutions. When a delay is present, spurious solutions are also found to exist within the linear stability limit, and this is seen to affect the dynamics. Potential symmetry in the problem is identified and it is shown that in certain circumstances the bifurcation patterns depend dramatically upon whether the initial data shares the symmetry.

KW - discretised evolution equation

KW - diffusion

KW - linear stability limit

KW - bifurcation patterns

KW - mathematics

UR - http://dx.doi.org/10.1016/0168-9274(95)00051-U

U2 - 10.1016/0168-9274(95)00051-U

DO - 10.1016/0168-9274(95)00051-U

M3 - Article

VL - 18

SP - 155

EP - 173

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 1-3

ER -