Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

Abstract

Abstract. We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [10] with a jump discontinuity 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 andFirstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity.This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.To derive these new effects in a mathematically rigorous way, we use analytical tools like LiapunovSchmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work [33].Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.1991 Mathematics Subject Classification. Primary 35B35, 76E30; Secondary 35B40, 76E06.

Pattern formation in reaction-diffusion systems with piecewise kinetic modulation: An example study of heterogeneous kinetics

“…However, the standard procedure of stability analysis is not easily extendable to the case with spatially dependent coefficients as, for example, a homogeneous steady state does not exist in general given spatially dependent kinetics. Probably the most well understood effect of heterogeneity comes from the shadow limit in Gierer-Meinhardt kinetics using spike solutions [11,35,33] but this requires the applicability of the shadow limit. Further the case of spatially dependent diffusion coefficient was analysed in [1] with a step function representing the dependency.…”

Mathematical Modeling of Plant Root Hair Initiation: Dynamics of Localized Patches

“…The focus in these studies is to characterize whether traveling fronts either penetrate through, are reflected from, or are pinned by the inhomogeneity. In Wei and Winter [53] the influence of a discontinuous inhibitor diffusion coefficient on the existence and stability of spikes in a Gierer-Meindhardt system is investigated. We are not aware of any study of the stability and dynamics of localized spikes when the spatial inhomogeneity arises in the nonlinear terms of the reaction kinetics.…”

Turing patterns resulting from a Sturm-Liouville problem

“…For instance, in all the above, it has been assumed that D u and D v are constant diffusion coefficients. Experimental evidence that in some biological systems spatial inhomogeneities are importan to regulate patterns lead to several generalizations of the reaction diffusion problem: when one of the diffusion coefficients either depends on the spatial variables [4,5] or is discontinuous [4,6]; time-dependent [7] or concentration-dependent diffusion coefficients [8][9][10]; spatially varying parameters [11,12]; reaction diffusion system in a chanel with the projected Fick-Jacobs-Zwanzig operator (with a diffusion coefficient that depends on the longitudinal coordinate) [13], as well as pattern formation with superdiffusion [14] or anomalous diffusion [15].…”