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

D.J. Higham, T. Sardar

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)155-173
Number of pages18
JournalApplied Numerical Mathematics
Volume18
Issue number1-3
DOIs
Publication statusPublished - 1995

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Nonlinear Reaction-diffusion Equations
Fixed point
Linear Stability
Long-time Behavior
Spurious Solutions
Symmetry
Population dynamics
Population Dynamics
Logistics
Spacing
Evolution Equation
Nonlinear Problem
Bifurcation
Numerical Solution
Grid
Term
Model

Keywords

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

Cite this

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

In: Applied Numerical Mathematics, Vol. 18, No. 1-3, 1995, p. 155-173.

Research output: Contribution to journalArticle

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AU - Sardar, T.

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