Spike solutions in Gierer-Meinhardt model with a time dependent anomaly exponent

Experimental evidence of complex dispersion regimes in natural systems, where the growth of the 7 mean square displacement in time cannot be characterised by a single power, has been accruing 8 for the past two decades. In such processes the exponent γ(t) in hr2i ∼ tγ(t) at times might be 9 approximated by a piecewise constant function, or it can be a continuous function. Variable order 10 differential equations are an emerging mathematical tool with a strong potential to model these 11 systems. However, variable order differential equations are not tractable by the classic differential 12 equations theory. This contribution illustrates how a classic method can be adapted to gain insight 13 into a system of this type. Herein a variable order Gierer-Meinhardt model is posed, a generic 14 reaction– diffusion system of a chemical origin. With a fixed order this system possesses a solution 15 in the form of a constellation of arbitrarily situated localised pulses, when the components’ diffu- 16 sivity ratio is asymptotically small. The pattern was shown to exist subject to multiple step-like 17 transitions between normal diffusion and sub-diffusion, as well as between distinct sub-diffusive 18 regimes. The analytical approximation obtained permits qualitative analysis of the impact thereof. 19 Numerical solution for typical cross-over scenarios revealed such features as earlier equilibration 20 and non-monotonic excursions before attainment of equilibrium. The method is general and allows 21 for an approximate numerical solution with any reasonably behaved γ(t).