EXISTENCE AND STABILITY OF FIXED-POINTS FOR A DISCRETIZED NONLINEAR REACTION-DIFFUSION EQUATION WITH DELAY

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.