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.

Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model

“…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.…”

Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case

“…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].…”

“…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]).…”

Effect of Different Shape of Periodic Forces on Chaotic Oscillation in Brusselator Chemical System

“…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.…”