Spikes for the Gierer-meinhardt System with Many Segments of Different Diffusivities

We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [5] with a large number of jump discontinuities in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini and Sherratt [1], [3], [9]. We first review results on the case of two segments given in [25], concerning one-spike steady states: the existence of interior spikes located away from the jump discontinuity was established, along with a necessary condition for the position of the spike, namely, the spike can be located in one-and-only-one of the two subintervals separated by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike does not hold for constant inhibitor diffusivities. Secondly, there also exist spikes whose distance from the jump discontinuity is of the same order as its spatial extent. It turns out that, generically, there either exist two different one-spike steady states near the jump discontinuity or there is none. In this paper, we prove a conjecture raised in [25]: We show that one of the spikes is stable while the other is unstable, using an eigenfunction constructed by outer and inner expansions. Moreover, since our argument involves only the two immediate segments around the jump discontinuity, the result holds for any number of segments. Next, we extend the interior spike results on the case of two segments (one jump) to an arbitrary number of segments. By analyzing the derivatives of the regular part of a Green’s function, we give a simple classification of interior segments according to the signs at both ends of the segment: There exists a stable spike, an unstable spike or there does not exist any spike in the segment which is away from the jump discontinuities. We also give explicit formulas of the solutions and conditions for existence for the case of three segments, which has one interior segment. Finally, we confirm our results by illustrating the long-term dynamical behavior of the system using numerical computations. We observe a moving spike which converges to a stationary interior spike, a spike near a jump discontinuity or a boundary spike. Dedicated to Professor Masayasu Mimura on the occasion of his 65th birthday