2009
DOI: 10.1007/s11464-009-0026-4
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Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models

Abstract: Recent advances in abstract local and global bifurcation theory is briefly reviewed. Several applications are included to illustrate the applications of abstract theory, and it includes Turing instability of chemical reactions, pattern formation in water limited ecosystems, and diffusive predator-prey models.

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Cited by 22 publications
(7 citation statements)
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“…Similar work along this line has been done recently for Gierer-Meinhardt model (Liu et al, 2010;Ruan, 1998), Sel'kov model (Han & Bao, 2009) and a biomolecular model with autocatalysis and saturation law (Yi et al, 2010). See also Shi (2009) for a recent survey on abstract bifurcation theorems and applications to spatiotemporal models from ecology and biochemistry.…”
Section: Introductionmentioning
confidence: 53%
“…Similar work along this line has been done recently for Gierer-Meinhardt model (Liu et al, 2010;Ruan, 1998), Sel'kov model (Han & Bao, 2009) and a biomolecular model with autocatalysis and saturation law (Yi et al, 2010). See also Shi (2009) for a recent survey on abstract bifurcation theorems and applications to spatiotemporal models from ecology and biochemistry.…”
Section: Introductionmentioning
confidence: 53%
“…The bifurcation of spatial nonhomogeneous steady state solutions from homogeneous ones is one of known mechanisms of pattern formation, hence it has been considered by many authors [1,[4][5][6][7]14,15,25,28,[37][38][39][40][41]. One famous example of bifurcations is the Turing bifurcation in which a diffusion coefficient is used as bifurcation parameter (see for example [14,28,37]), but recent studies show that other parameters can also generate bifurcations when there is no restriction on the diffusion coefficients (see [15,41]). The global properties of the bifurcating branches have also been considered (see [1,4,6,38]), following the celebrated global bifurcation theorem of Rabinowitz [34].…”
Section: Global Bifurcations In Diffusive Predator-prey Systemsmentioning
confidence: 99%
“…One famous example of bifurcations is the Turing bifurcation in which a diffusion coefficient is used as bifurcation parameter (see for example [14,28,37]), but recent studies show that other parameters can also generate bifurcations when there is no restriction on the diffusion coefficients (see [15,41]). The global properties of the bifurcating branches have also been considered (see [1,4,6,38]), following the celebrated global bifurcation theorem of Rabinowitz [34]. In particular, it was shown that in some cases, the branches of non-trivial steady state solutions are unbounded (see [14,15,28]).…”
Section: Global Bifurcations In Diffusive Predator-prey Systemsmentioning
confidence: 99%
“…The study of nonlinear chemical dynamics (NCD) has flourished in the past three decades [1][2][3][4][5][6][7][8][9][10]. The prototypical phenomenon of NCD is chemical oscillation.…”
Section: Introductionmentioning
confidence: 99%