Stability Analysis of Reaction-Diffusion Models with Delayed Reaction Kinetics
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The linear stability of localized spike solutions to the one-dimensional Gierer-Meinhardt activator-inhibitor model with delayed nonlinear reaction kinetics is analyzed both analytically and numerically. In the limit of slow activator diffusivity, we show that delay destabilizes the equilibrium solution, and we find critical values at which a Hopf bifurcation is observed in both the spike position and amplitude. For specific cases of delayed reaction kinetics, we formulate the nonlocal eigenvalue problem and we study the stability of both the small and large eigenvalues. For the small eigenvalues, we show that in some cases the reduced system of ordinary differential equations, for the motion of the slow evolving spikes, undergoes a Hopf bifurcation. Instabilities in the spike profile are also considered, and we show that the equilibrium solution is unstable as delay is increased beyond a critical Hopf bifurcation value. For one-spike solutions, we find that instability in the profile is triggered before the positional instability, except in the case where the degradation of activator is delayed where stable positional oscillations are observed. The analytical results are validated using numerical simulations. In addition, we study an example of quorum sensing behaviour modelled by a two-dimensional cell-bulk model coupled to delayed intracellular dynamics. In this model, the essential process of cell-to-cell communication is achieved by the diffusion of a signalling molecule in a well-mixed bulk medium between spatially segregated active cells. Assuming a very large diffusion limit, we investigate the onset of oscillatory instabilities due to coupling with delayed intracellular dynamics. The cell-bulk model, for the case of a single active cell containing one intracellular species, is reduced to a finite system of nonlinear delay ordinary differential equations and studied both analytically and numerically. Using Hill function-type intracellular kinetics with fixed delay, we show that delayed cell-bulk coupling triggers sustained oscillations as delay increases beyond the critical Hopf bifurcation threshold.