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Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model. (English) Zbl 1203.35037
Summary: We have investigated a homogeneous reaction-diffusion bimolecular model with autocatalysis and saturation law subject to Neumann boundary conditions. We mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive equilibrium solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbits and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive biomolecular system. Numerical examples are presented to support our theoretical analysis.

MSC:
35B36 Pattern formations in context of PDEs
35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
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[1] Blat, J.; Brown, K.J., Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. math. anal., 17, 1339-1353, (1986) · Zbl 0613.35008
[2] Bonilla, L.L.; Velarde, M.G., Singular perturbations approach to the limit cycle and global patterns in a nonlinear diffusion – reaction problem with autocatalysis ans saturation law, J. math. phys., 20, 2692-2703, (1979) · Zbl 0455.35107
[3] Du, Y.-H., Uniqueness, multiplicity and stability for positive solutions of a pair of reaction – diffusion equations, Proc. roy. soc. Edinburgh, 126 A, 777-809, (1996) · Zbl 0864.35035
[4] Ibanez, J.L.; Velarde, M.G., Multiple steady states in a simple reaction – diffusion model with michaelis – menten (first-order hinshelwood-Langmuir) saturation law: the limit of large separation in the two diffsion constants, J. math. phys., 19, 151-156, (1978)
[5] Jang, J.; Ni, W.M.; Tang, M.X., Global bifurcation and structure of Turing patterns in the 1-D lengyel – epstein model, J. dyn. differ. equ., 16, 297-320, (2004) · Zbl 1072.35091
[6] J.-Y. Jin, J.-P. Shi, J.-J. Wei, F.-Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction (in press). · Zbl 1288.35051
[7] J.-X. Liu, F.-Q. Yi, J.-J. Wei, Mutiple bifurcations and pattern formation in diffusive Geier-Minhardt model, Int. J. Bifurcation Chaos (in press).
[8] Peng, R.; Shi, J.-P., Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case, J. differ. equ., 247, 3, 866-886, (2009) · Zbl 1169.35328
[9] Peng, R.; Shi, J.-P.; Wang, M.-X., Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. appl. math., 67, 5, 1479-1503, (2007) · Zbl 1210.35268
[10] Ruan, W., Asymptotic behavior and positive steady-state solutions of a reaction – diffusion model with autocatalysis and saturation law, Nonlinear anal. TMA, 21, 439-456, (1993) · Zbl 0796.35074
[11] Yi, F.-Q.; Wei, J.-J.; Shi, J.-P., Diffusion-driven instability and bifurcation in the lengyel – epstein system, Nonlinear anal. RWA, 9, 3, 1038-1051, (2008) · Zbl 1146.35384
[12] Yi, F.-Q.; Wei, J.-J.; Shi, J.-P., Bifurcation and spatio-temporal patterns in a diffusive homogenous predator – prey system, J. differ. equ., 246, 5, 1944-1977, (2009) · Zbl 1203.35030
[13] Nishiura, Y., Global structure of bifurcating solutions of some reaction – diffusion systems, SIAM J. math. anal., 13, 555-593, (1982) · Zbl 0501.35010
[14] Shi, J.-P., Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Front. math. China, 4, 407-424, (2009) · Zbl 1176.35021
[15] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[16] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044
[17] Dancer, E.N., On the indices of fixed points of mappings in cones and applications, J. math. anal. appl., 91, 131-151, (1983) · Zbl 0512.47045
[18] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. ration. mech. anal., 52, 161-180, (1973) · Zbl 0275.47044
[19] Peng, R.; Shi, J.-P.; Wang, M.-X., On stationary patterns of a reaction – diffusion model with autocatalysis and saturation law, Nonlinearity, 21, 7, 1471-1488, (2008) · Zbl 1148.35094
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