On global bifurcation for quasilinear elliptic systems on bounded domains

General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of [Jacobo Pejsachowicz, Patrick J. Rabier, Degree theory for C(1) Fredholm mappings of index 0, J. Anal. Math. 76 (1998) 289-319] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied. (C) 2008 Elsevier Inc. All rights reserved

Classification of four-body central configurations with three equal masses

It is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior. We use analytic methods to show that besides the family of equilateral triangle configurations, there are exactly one family of concave and one family of convex central configurations, which completely classifies such central configurations. (C)2009 Elsevier Inc. All rights reserved

Global Continuum and Multiple Positive Solutions to a P-Laplacian Boundary-Value Problem

A p-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from (lambda, u) = (0, 0) is shown, and it can be extended to lambda = infinity. Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any lambda \u3e 0; thus the continuum C is indeed a continuous curve globally defined for all lambda \u3e 0. In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity

Relaxation Oscillation Profile of Limit Cycle in Predator-Prey System

It is known that some predator-prey system can possess a unique limit cycle which is globally asymptotically stable. For a prototypical predator-prey system, we show that the solution curve of the limit cycle exhibits temporal patterns of a relaxation oscillator, or a Heaviside function, when certain parameter is small

The Role of Higher Vorticity Moments in a Variational Formulation of Barotropic Flows on a Rotating Sphere

The effects of a higher vorticity moment on a variational problem for barotropic vorticity on a rotating sphere is examined rigorously in the framework of the Direct Method. This variational model differs from previous work on the Barotropic Vorticity Equation (BVE) in relaxing the angular momentum constraint, which then allows us to state and prove theorems that give necessary and sufficient conditions for the existence and stability of constrained energy extremals in the form of super and sub-rotating solid-body steady flows. Relaxation of angular momentum is a necessary step in the modeling of the important tilt instability where the rotational axis of the barotropic atmosphere tilts away from the fixed north-south axis of planetary spin. These conditions on a minimal set of parameters consisting of the planetary spin, relative enstrophy and the fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows

Global stability in a diffusive Holling-Tanner predator-prey model

A diffusive Holling-Tanner predator-prey model with no-flux boundary condition is considered, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition. (C) 2011 Elsevier Ltd. All rights reserved

Existence of positive solutions to Kirchhoff type problems with zero mass

The existence of positive solutions depending on a nonnegative parameter lambda to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais-Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais-Smale sequences for the variable-coefficient case. (C) 2013 Elsevier Inc. All rights reserved

Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system

The Lengyel-Epstein reaction-diffusion system of the CIMA reaction is revisited. We construct a Lyapunov function to show that the constant equilibrium solution is globally asymptotically stable when the feeding rate of iodide is small. We also show that for small spatial domains, all solutions eventually converge to a spatially homogeneous and time-periodic solution. (C) 2008 Elsevier Ltd. All rights reserved

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation. (c) 2008 Elsevier Inc. All rights reserved