Hopf bifurcation in spatially homogeneous and inhomogeneous autocatalysis models

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Abstract

This paper is concerned with spatially homogeneous and inhomogeneous autocatalysis models with arbitrary order. For the spatially homogeneous model, the existence and stability of Hopf bifurcation surrounding the interior equilibrium are considered. For the model subject to zero-flux boundary conditions, Turing instability of the interior equilibrium is firstly discussed and Turing instability region regarding the diffusion coefficients is established. Then by the center manifold theory and normal form method, the existence of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained, which is much more difficult and complicated than dealing with the spatially homogeneous case. Finally, to verify our analytical results, some examples of numerical simulations are also included.

Keywords

Autocatalysis model
Hopf bifurcation
Stability

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The work is supported by the National Natural Science Foundation of China (Nos. 11271236, 11001160), the Natural Science Foundation of Shaanxi Province (No. 2011JQ1015), and the Foundations of Shaanxi Educational Committee (No. 12JK0856) and SUST (No. 2012SB001).