Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of Cima chemical reactions

Bifurcations of spatially nonhomogeneous periodic solutions and steady state solutions are rigorously proved for a reaction-diffusion system modeling CIMA chemical reaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics including Turing instability and oscillatory behavior. Examples of numerical simulation are also shown to support and strengthen the analytical approach.

[1]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[2]  Abraham Trembley.  Mémoires, pour servir à l'histoire d'un genre de polypes d'eau douce, à bras en forme de cornes , 1975 .

[3]  西浦 廉政,et al.  Global structure of bifurcating solutions of some reaction-diffusion systems , 1982 .

[4]  Izumi Takagi,et al.  Point-condensation for a reaction-diffusion system , 1986 .

[5]  H. Lenhoff,et al.  Hydra and the Birth of Experimental Biology, 1744: Abraham Trembley's Memoires Concerning the Polyps , 1986 .

[6]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[7]  I. Epstein,et al.  Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System , 1991, Science.

[8]  J. Boissonade,et al.  Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction , 1991 .

[9]  I. Epstein,et al.  A chemical approach to designing Turing patterns in reaction-diffusion systems. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[10]  David J. Wollkind,et al.  Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems , 1995 .

[11]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[12]  P. Maini,et al.  Spatial pattern formation in chemical and biological systems , 1997 .

[13]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[14]  I. Epstein,et al.  An Introduction to Nonlinear Chemical Dynamics , 1998 .

[15]  Stephen L. Judd,et al.  Simple and superlattice Turing patterns in reaction-diffusion systems: bifurcation, bistability, and parameter collapse , 1998, patt-sol/9807002.

[16]  Junping Shi,et al.  Persistence and Bifurcation of Degenerate Solutions , 1999 .

[17]  David J. Wollkind,et al.  Chemical Turing Pattern Formation Analyses: Comparison of Theory with Experiment , 2000, SIAM J. Appl. Math..

[18]  James D. Murray,et al.  Spatial models and biomedical applications , 2003 .

[19]  Jaeduck Jang,et al.  Global Bifurcation and Structure of Turing Patterns in the 1-D Lengyel–Epstein Model , 2004 .

[20]  Wei-Ming Ni,et al.  Turing patterns in the Lengyel-Epstein system for the CIMA reaction , 2005 .

[21]  Junjie Wei,et al.  Diffusion-driven instability and bifurcation in the Lengyel-Epstein system , 2008 .

[22]  Sze-Bi Hsu,et al.  Relaxation , 2021, Graduate Studies in Mathematics.

[23]  Junping Shi,et al.  On global bifurcation for quasilinear elliptic systems on bounded domains , 2009 .

[24]  Junjie Wei,et al.  Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system , 2009, Appl. Math. Lett..