Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models

Frontiers of Mathematics in China

Volumn 4, Issue 3, 2009, Pages 407-424

  Shi, Junping ^a,b  

^a Dept of Economics and Thomas Jefferson Program in Public Policy at the College of William and Mary   (United States)

^b HARBIN NORMAL UNIVERSITY   (China)

Author keywords

Bifurcation; Reaction diffusion model

Indexed keywords

EID: 67650885603     PISSN: 16733452     EISSN: 16733576     Source Type: Journal    
DOI: 10.1007/s11464-009-0026-4     Document Type: Article

References (46)

1. Ambrosetti A, Prodi G. A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, Vol 34. Cambridge: Cambridge University Press, 1995
2. Cantrell R S, Cosner C. Spatial Ecology via Reaction-diffusion Equation. Wiley Series in Mathematical and Computational Biology. New York: John Wiley & Sons Ltd, 2003
3. Chang K -C. Methods in Nonlinear Analysis. Springer Monographs in Mathematics. Berlin: Springer-Verlag, 2005
4. K. S. Cheng 1981 Uniqueness of a limit cycle for a predator-prey system SIAM J Math Anal 12 4 541 548
5. Chow S N, Hale J K. Methods of Bifurcation Theory. New York-Berlin: Springer-Verlag, 1982
6. Conway E D. Diffusion and predator-prey interaction: pattern in closed systems. In: Fitzgibbon W E, ed. Partial Differential Equations and Dynamical Systems. Res Notes in Math, Vol 101. Boston-London: Pitman, 1984, 85-133
7. M. G. Crandall P. H. Rabinowitz 1971 Bifurcation from simple eigenvalues Jour Func Anal 8 321 340
8. M. G. Crandall P. H. Rabinowitz 1973 Bifurcation, perturbation of simple eigenvalues and linearized stability Arch Rational Mech Anal 52 161 180
9. P. De Kepper V. Castets E. Dulos J. Boissonade 1991 Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction Physica D 49 161 169
10. Deimling K. Nonlinear Functional Analysis. Berlin-New York: Springer-Verlag, 1985
11. Du Y H, Shi J P. Some recent results on diffusive predator-prey models in spatially heterogeneous environment. In: Brunner H, Zhao X Q, Zou X F, eds. Nonlinear Dynamics and Evolution Equations. Fields Institute Communications, 48. Providence: American Mathematical Society, 2006, 95-135
12. Epstein I R, Pojman J A. An Introduction to Nonlinear Chemical Dynamics. Oxford: Oxford University Press, 1998
13. Gray P, Scott S K. Chemical Oscillations and Instabilities: Nonlinear Chemical Kinetics. Oxford: Clarendon Press, 1990
14. S. M. Han H. Benaroya T. Wei 1999 Dynamics of transversely vibrating beams using four engineering theories Jour Sound and Vibr 225 5 935 988 (Pubitemid 129536982)
15. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol 840. Berlin-New York: Springer-Verlag, 1981
16. Hsu S -B. Ordinary Differential Equations with Applications. Series on Applied Mathematics, 16. Hackensack: World Scientific Publishing Co Pte Ltd, 2006
17. Hsu S-B, Shi J P. Relaxation oscillator profile of limit cycle in predator-prey system. Disc Cont Dyns Syst -B (to appear)
18. J. Jang W. -M. Ni M. X. Tang 2004 Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model J Dynam Differential Equations 16 2 297 320 (Pubitemid 39519632)
19. Jiang J F, Shi J P. Bistability dynamics in some structured ecological models. In: Spatial Ecology: A Collection of Essays. Boca Raton: CRC Press, 2009 (in press)
20. Kielhöfer H. Bifurcation Theory. An Introduction with Applications to PDEs. Applied Mathematical Sciences, Vol 156. New York: Springer-Verlag, 2004
21. I. Lengyel I. R. Epstein 1991 Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system Science 251 650 652 (Pubitemid 21916886)
22. I. Lengyel I. R. Epstein 1992 A chemical approach to designing Turing patterns in reaction-diffusion system Proc Natl Acad Sci USA 89 3977 3979
23. P. Liu J. P. Shi Y. W. Wang 2007 Imperfect transcritical and pitchfork bifurcations Jour Func Anal 251 2 573 600
24. López-Gómez J. Spectral Theory and Nonlinear Functional Analysis. Chapman & Hall/CRC Research Notes in Mathematics, Vol 426. Boca Raton: Chapman & Hall/CRC, 2001
25. A. B. Medvinsky S. V. Petrovskii I. A. Tikhonova H. Malchow B. -L. Li 2002 Spatiotemporal complexity of plankton and fish dynamics SIAM Rev 44 3 311 370 (Pubitemid 35048789)
26. E. Meron E. Gilad J. von Hardenberg M. Shachak Y. Zarmi 2004 Vegetation patterns along a rainfall gradient Chaos Solitons Fractals 19 367 376
27. Murray J D. Mathematical Biology. I. An Introduction. 3rd Ed. Interdisciplinary Applied Mathematics, Vol 17. New York: Springer-Verlag, 2003
28. Murray J D. Mathematical Biology. II. Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, Vol 18. New York: Springer-Verlag, 2003
29. W. -M. Ni M. X. Tang 2005 Turing patterns in the Lengyel-Epstein system for the CIMA reaction Trans Amer Math Soc 357 10 3953 3969
30. Nirenberg L. Topics in Nonlinear Functional Analysis. Courant Lecture Notes in Mathematics, Vol 6. New York University, Courant Institute of Mathematical Sciences, New York. Providence: American Mathematical Society, 2001
31. Y. Nishiura 1982 Global structure of bifurcating solutions of some reaction-diffusion systems SIAM J Math Anal 13 4 555 593
32. Okubo A, Levin S. Diffusion and Ecological Problems: Modern Perspectives. 2nd Ed. Interdisciplinary Applied Mathematics, Vol 14. New York: Springer-Verlag, 2001
33. 1 Fredholm mappings of index 0 J Anal Math 76 289 319 (Pubitemid 128438681)
34. P. H. Rabinowitz 1971 Some global results for nonlinear eigenvalue problems Jour Func Anal 7 487 513
35. Reiss E L. Column buckling-an elementary example of bifurcation. In: Keller J B, Antman S, eds. Bifurcation Theory and Nonlinear Eigenvalue Problems. New York-Amsterdam: W A Benjamin, Inc, 1969, 1-16
36. Reiss E L. Imperfect bifurcation. In: Applications of Bifurcation Theory (Proc Advanced Sem, Univ Wisconsin, Madison, Wis, 1976). Publ Math Res Center Univ Wisconsin, No 38. New York: Academic Press, 1977, 37-71
37. M. L. Rosenzweig 1971 Paradox of enrichment: destabilization of exploitation ecosystems in ecological time Science 171 3969 385 387
38. J. P. Shi 1999 Persistence and bifurcation of degenerate solutions Jour Func Anal 169 2 494 531
39. Shi J P. Solution Set of Semilinear Elliptic Equations: Global Bifurcation and Exact Multiplicity. Singapore: World Scientific Publishing Co Pte Ltd, 2009
40. J. P. Shi X. F. Wang 2009 On global bifurcation for quasilinear elliptic systems on bounded domains Jour Diff Equations 246 7 2788 2812
41. Timosehnko S P. History of Strength of Materials. New York: Dover Publications, Inc, 1953
42. A. M. Turing 1952 The chemical basis of morphogenesis Philosophical Transaction of Royal Society of London B237 37 72
43. von Hardenberg J, Meron E, Shachak M, Zarmi Y. Diversity of vegetation patterns and desertification. Phys Rev Lett, 2001, 87: 198101
44. F. Q. Yi J. J. Wei J. P. Shi 2008 Diffusion-driven instability and bifurcation in the Lengyel-Epstein system Nonlinear Anal Real World Appl 9 3 1038 1051
45. F. Q. Yi J. J. Wei J. P. Shi 2009 Bifurcation and spatio-temporal patterns in a diffusive homogenous predator-prey system Jour Diff Equations 246 5 1944 1977
46. F. Q. Yi J. J. Wei J. P. Shi 2009 Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system Appl Math Lett 22 1 52 55