Dynamics and stability of spike-type solutions to a one dimensional Gierer–Meinhardt model with sub-diffusion

Abstract

The dynamics and stability of spike-type patterns to a sub-diffusive Gierer–Meinhardt reaction–diffusion system is studied in a one dimensional spatial domain. A differential algebraic system (DAE) is derived to characterise the dynamics of an n-spike quasi-equilibrium pattern in the presence of sub-diffusion. With sub-diffusive effects it is shown that quasi-equilibrium spike patterns exist for diffusivity ratios asymptotically smaller than for the case of regular diffusion, and that the spikes approach their equilibrium locations at an algebraic, rather than exponential, rate in time.A new non-local eigenvalue problem (NLEP) is derived to examine the stability of an n-spike pattern. For a two spike pattern sub-diffusion has little effect on the competition instability threshold, whereas the threshold associated with an oscillatory instability of the spike profile increases significantly. Furthermore, for a multi-spike pattern it is shown that an asynchronous oscillatory instability of the spike profile, rather than a synchronous oscillatory instability characteristic of the case of regular diffusion, is the dominant instability when the anomaly index γ is below a certain threshold. Detailed numerical results are presented for the two spike case.