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**Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay.**
*(English)*
Zbl 0834.65079

The long time behaviour of a discretized 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. As observed elsewhere, increasing the time step beyond the linear stability limit may induce stable spurious steady states. When a delay is present, spurious solutions are also found to exist within the linear stability limit and this affects the dynamics.

Potential symmetry in the problem is identified. It is shown that in certain circumstances the bifurcation patterns depend dramatically upon whether the initial data shares the symmetry.

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. As observed elsewhere, increasing the time step beyond the linear stability limit may induce stable spurious steady states. When a delay is present, spurious solutions are also found to exist within the linear stability limit and this affects the dynamics.

Potential symmetry in the problem is identified. It is shown that in certain circumstances the bifurcation patterns depend dramatically upon whether the initial data shares the symmetry.

Reviewer: S.L.Campbell (Raleigh)

### MSC:

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

92D25 | Population dynamics (general) |

35K57 | Reaction-diffusion equations |

34K05 | General theory of functional-differential equations |

### Keywords:

fixed points; nonlinear reaction-diffusion equation; evolution equation; population dynamics; stability; delay; bifurcation
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\textit{D. J. Higham} and \textit{T. Sardar}, Appl. Numer. Math. 18, No. 1--3, 155--173 (1995; Zbl 0834.65079)

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