STABILITY OF SPIKY SOLUTIONS IN A REACTION-DIFFUSION SYSTEM WITH FOUR MORPHOGENS ON THE REAL LINE
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AbstractWe study a reaction-diffusion system with four morphogens which has been suggested in [H. Takagi and K. Kaneko, Europhys. Lett., 56 (2001), pp. 145-151]. This system is a generalization of the Gray-Scott model [P. Gray and S. K. Scott, Chem. Eng. Sci., 38 (1983), pp. 29-43; 39 (1984), pp. 1087-1097] and allows for multiple activators and multiple substrates. We construct single-spike solutions on the real line and establish their stability properties in terms of conditions of connection matrices which describe the interaction of the components. We use a rigorous analysis for the linearized operator around single-spike solutions based on nonlocal eigenvalue problems and generalized hypergeometric functions. The following results are established for two activators and two substrates: Spiky solutions may be stable or unstable, depending on the type and strength of the interaction of the morphogens. In particular, it is shown that these patterns are stabilized in the following two cases. Case 1: interaction of different activators with each other (off-diagonal interaction of activators). Case 2: variation in strength of interaction of activators with different substrates (e. g., each activator has its preferred substrate).
All Author(s) ListWei JC, Winter M
Journal nameSIAM Journal on Mathematical Analysis
Year2010
Month1
Day1
Volume Number42
Issue Number6
PublisherSIAM PUBLICATIONS
Pages2818 - 2841
ISSN0036-1410
eISSN1095-7154
LanguagesEnglish-United Kingdom
Keywordsfour morphogens; pattern formation; reaction-diffusion system; spike solutions; stability
Web of Science Subject CategoriesMathematics; Mathematics, Applied; MATHEMATICS, APPLIED

Last updated on 2020-24-10 at 00:54