Mutually Exclusive Spiky Pattern and Segmentation Modeled by the Five-Component Meinhardt--Gierer System
نویسندگان
چکیده
We consider the five-component Meinhardt-Gierer model for mutually exclusive patterns and segmentation which was proposed in [11]. We prove rigorous results on the existence and stability of mutually exclusive spikes which are located in different positions for the two activators. Sufficient conditions for existence and stability are derived, which depend in particular on the relative size of the various diffusion constants. Our main analytical methods are the Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are confirmed by numerical simulations.
منابع مشابه
Existence and Stability Analysis of Spiky Solutions for the Gierer-meinhardt System with Large Reaction Rates
We study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either...
متن کاملIdentification of Space-Time Distributed Parameters in the Gierer-Meinhardt Reaction-Diffusion System
We consider parameter identification for the classic Gierer-Meinhardt reactiondiffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39] was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology. In our paper the parameters are extended in time and space a...
متن کاملOn Single Interior Spike Solutions of Gierer-meinhardt System: Uniqueness and Spectrum Estimates
We study the interior spike solutions to a steady state problem of the shadow system of the Gierer-Meinhardt system arising from biological pattern formation. We rst show that at a nondegenerate peak point the interior spike solution is locally unique and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow s...
متن کاملOn the Gierer-meinhardt System with Precursors
We consider the following Gierer-Meinhardt system with a precursor μ(x) for the activator A in R: At = 2A ′′ − μ(x)A + A2 H in (−1, 1), τHt = DH ′′ −H + A in (−1, 1), A′(−1) = A′(1) = H ′(−1) = H ′(1) = 0. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stabili...
متن کاملThe Identification of Space-time Distributed Parameters in Reaction-diffusion Systems
We consider a general methodology for parameter identification in systems of reaction-diffusion equations. To demonstrate the method we focus on the classic Gierer-Meinhardt reaction-diffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39] was formulated with constant parameters and has been used as a prototype system for investigating...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM Journal of Applied Mathematics
دوره 69 شماره
صفحات -
تاریخ انتشار 2008